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<span class="kw">let </span><font color="Maroon" title="c1">BL</font> be  non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  <a href="lattices.html#NM8" title="LATTICES:NM.8">B_Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c1">BL</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <span class="p1">(<span class="default"><a href="lopclset.html#K11" title="LOPCLSET:func.11">StoneSpace</a> <font color="Maroon" title="c1">BL</font></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">X</font> = <font color="Maroon" title="c2">x</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K11" title="LOPCLSET:func.11">StoneSpace</a> <font color="Maroon" title="c1">BL</font></span>)</span> ;<br/>
<a NAME="E3:33"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c3">X</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="ref" href="lopclset.html#T2" target="_self" title="LOPCLSET:th.2">Th2</a></span>;<br/>
<a NAME="E4:33"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<font color="Maroon" title="c3">X</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopclset.html#T3" target="_self" title="LOPCLSET:th.3">Th3</a></span>;<br/>
<a NAME="E5:33"/>
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K11" title="LOPCLSET:func.11">StoneSpace</a> <font color="Maroon" title="c1">BL</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E6:33"/><span class="kw">then </span>
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c1">BL</font></span>)</span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopclset.html#D8" target="_self" title="LOPCLSET:def.8">Def8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">B</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c1">BL</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E8:33"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">and </span><br/><a NAME="E9:33"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font>
 ;<br/>
<a NAME="E10:33"/>
<font color="Maroon" title="c3">X</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="#E4:33"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></span>;<br/>
<a NAME="E11:33"/><span class="kw">then </span>
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K11" title="LOPCLSET:func.11">StoneSpace</a> <font color="Maroon" title="c1">BL</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E12:33"/><span class="kw">then </span>
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c1">BL</font></span>)</span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopclset.html#D8" target="_self" title="LOPCLSET:def.8">Def8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">C</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c1">BL</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E14:33"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">and </span><br/><a NAME="E15:33"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
<font color="Maroon" title="c5">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font>
 ;<br/>
<a NAME="E16:33"/>
<font color="Maroon" title="c4">B</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c5">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:33"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E15:33"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E17:33"/><span class="kw">then </span>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c5">C</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a></span>;<br/>
<a NAME="E18:33"/><span class="kw">then </span>
<font color="Maroon" title="c3">X</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</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>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E14:33"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</a></span>;<br/>
<a NAME="E19:33"/><span class="kw">then </span>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="lopclset.html#K11" title="LOPCLSET:func.11">StoneSpace</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T18" title="PRE_TOPC:th.18">PRE_TOPC:18</a></span>;<br/>
<a NAME="E20:33"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A7</font></span>: 
 <a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c1">BL</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopclset.html#D8" target="_self" title="LOPCLSET:def.8">Def8</a></span>;<br/>
<a NAME="E21:33"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p2">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a></span>;<br/>
<a NAME="E22:33"/><span class="kw">then </span>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c1">BL</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T99" title="ZFMISC_1:th.99">ZFMISC_1:99</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:33"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c1">BL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:33"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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