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

<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="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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></span><br/>

<a NAME="E1:45"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
 <a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="lopclset.html#E49"><span class="lab"><font color="Green" title="E26">Lm2</font></span></a></span>;<br/>
<a NAME="E2:45"/>
 <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> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:45_1"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font></span><br/>

<span class="kw">then </span><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="E3:45_1"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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/>
<a NAME="E4:45_1"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
( <font color="Maroon" title="c3">X</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <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="txt" href="#E1:45_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="lopclset.html#T2" target="_self" title="LOPCLSET:th.2">Th2</a>, <a class="ref" href="lopclset.html#T3" target="_self" title="LOPCLSET:th.3">Th3</a></span>;<br/>
<a NAME="E5:45_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A5</font></span>: 
( <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>  &amp; <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#V4" title="PRE_TOPC:attr.4">closed</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a>, <a class="ref" href="tops_1.html#T30" title="TOPS_1:th.30">TOPS_1:30</a></span>;<br/>
<a NAME="E6:45_1"/>
<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="#E4:45_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">A1</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="E8:45_1"/><span class="lab"><font color="Green" title="E33">A6</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">A1</font>
 <span class="kw">and </span><br/><a NAME="E9:45_1"/><span class="lab"><font color="Green" title="E34">A7</font></span>: 
<font color="Maroon" title="c4">A1</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="#E3:45_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a></span>;<br/>
<a NAME="E10:45_1"/>
<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="txt" href="#E5:45_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">A2</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="E12:45_1"/><span class="lab"><font color="Green" title="E35">A8</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">A2</font>
 <span class="kw">and </span><br/><a NAME="E13:45_1"/><span class="lab"><font color="Green" title="E36">A9</font></span>: 
<font color="Maroon" title="c5">A2</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="#E3:45_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a></span>;<br/>
<a NAME="E14:45_1"/><span class="lab"><font color="Green" title="E37">A10</font></span>: 
<font color="Maroon" title="c3">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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">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">set </span><font color="Maroon" title="c6">AA</font> = <font color="Maroon" title="c4">A1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c5">A2</font>;<br/>
<span class="lab"><font color="Green" title="E38">A11</font></span>:  <a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c1">BL</font> = 
 <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>
<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/>.= 
<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>
<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 href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c5">A2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:45_1"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E12:45_1"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</a></span>
;<br/>
<a NAME="E16:45_1"/><span class="lab"><font color="Green" title="E39">A12</font></span>: 
<font color="Maroon" title="c4">A1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c5">A2</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="#E9:45_1"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E13:45_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">Y</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">A1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c5">A2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E18:45_1"/><span class="lab"><font color="Green" title="E40">A13</font></span>: 
 <a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c1">BL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c7">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:45_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="ref" href="lopclset.html#T38" target="_self" title="LOPCLSET:th.38">Th38</a></span>;<br/>
<a NAME="E19:45_1"/><span class="lab"><font color="Green" title="E41">A14</font></span>: 
<font color="Maroon" title="c7">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c5">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></span>;<br/>
<a NAME="E20:45_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A15</font></span>: 
<font color="Maroon" title="c7">Y</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="#E16:45_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">D</font> = <font color="Maroon" title="c4">A1</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c7">Y</font>;<br/>
<span class="lab"><font color="Green" title="E43">A16</font></span>: <font color="Maroon" title="c7">Y</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">A1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c5">A2</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c7">Y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:45_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">A1</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c7">Y</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A2</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c7">Y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A2</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c7">Y</font> implies <font color="Maroon" title="c9">y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">X</font> )</span><br/><span class="kw">assume </span><a NAME="E1:45_1_3"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A2</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c7">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">X</font></span><br/><a NAME="E2:45_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:45_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A17</font></span>: 
<font color="Maroon" title="c9">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:45_1"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/><a NAME="E4:45_1_3"/>
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/><a NAME="E5:45_1_3"/><span class="kw">then </span>
<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="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E6:45_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_1_3"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></span>;<br/><span class="kw">hence </span><a NAME="E7:45_1_3"/>
<font color="Maroon" title="c9">y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</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/></div><span class="kw">end;</span></div>
<a NAME="E23:45_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A18</font></span>: 
<font color="Maroon" title="c3">X</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A1</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c7">Y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E18:45_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E21:45_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="ref" href="setfam_1.html#T57" title="SETFAM_1:th.57">SETFAM_1:57</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_4"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:45_1_4"/>
( <font color="Maroon" title="c4">A1</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c4">A1</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c7">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_4_1"><span class="kw">suppose </span></a><a NAME="E1:45_1_4_1"/>
<font color="Maroon" title="c4">A1</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font></span><br/><div class="add"><a NAME="E2:45_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A19</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:45_1"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></span>;<br/><a NAME="E3:45_1_4_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">BL</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopclset.html#T36" target="_self" title="LOPCLSET:th.36">Th36</a></span>;<br/><span class="kw">hence </span><a NAME="E4:45_1_4_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="#E2:45_1_4_1"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_4_2"><span class="kw">suppose </span></a><a NAME="E1:45_1_4_2"/><span class="lab"><font color="Green" title="E45">A20</font></span>: 
<font color="Maroon" title="c4">A1</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c7">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font></span><br/><div class="add"><a NAME="E2:45_1_4_2"/><span class="lab"><font color="Green" title="E46">A21</font></span>: 
<font color="Maroon" title="c4">A1</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c7">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">D</font> = <font color="Maroon" title="c4">A1</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c7">Y</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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">by</span> <span class="lab"><a class="txt" href="#E1:45_1_4_2"><span class="lab"><font color="Green" title="E45">A20</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">D</font> = <font color="Maroon" title="c9">D</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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> ;<br/><a NAME="E5:45_1_4_2"/><span class="lab"><font color="Green" title="E47">A22</font></span>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">D</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">A1</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c7">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T97" title="ZFMISC_1:th.97">ZFMISC_1:97</a></span>;<br/><a NAME="E6:45_1_4_2"/>
<span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">A1</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c7">Y</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> <font color="Maroon" title="c4">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/><a NAME="E7:45_1_4_2"/><span class="kw">then </span>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:45_1"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E5:45_1_4_2"><span class="lab"><font color="Green" title="E47">A22</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E8:45_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A23</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:45_1"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><a NAME="E9:45_1_4_2"/><span class="lab"><font color="Green" title="E49">A24</font></span>: 
<font color="Maroon" title="c10">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:45_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E2:45_1_4_2"><span class="lab"><font color="Green" title="E46">A21</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E10:45_1_4_2"/><span class="lab"><font color="Green" title="E50">A25</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E11:45_1_4_2"/><span class="lab"><font color="Green" title="E51">A26</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span>
 ;<br/><a NAME="E12:45_1_4_2"/><span class="lab"><font color="Green" title="E52">A27</font></span>: 
 <a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopclset.html#T29" target="_self" title="LOPCLSET:th.29">Th29</a></span>;<br/><a NAME="E13:45_1_4_2"/>
 <a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T80" title="RELAT_1:th.80">RELAT_1:80</a></span>;<br/><a NAME="E14:45_1_4_2"/><span class="kw">then </span>
 <a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font>, <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:45_1_4_2"><span class="lab"><font color="Green" title="E50">A25</font></span></a>, <a class="txt" href="#E11:45_1_4_2"><span class="lab"><font color="Green" title="E51">A26</font></span></a>, <a class="txt" href="#E12:45_1_4_2"><span class="lab"><font color="Green" title="E52">A27</font></span></a>, <a class="ref" href="classes1.html#T85" title="CLASSES1:th.85">CLASSES1:85</a></span>;<br/><span class="kw">then </span> <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c10">D</font></span>)</span> = 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c10">D</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="classes1.html#T86" title="CLASSES1:th.86">CLASSES1:86</a></span>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c10">D</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_1_4_2"><span class="lab"><font color="Green" title="E49">A24</font></span></a>, <a class="ref" href="funct_2.html#T171" title="FUNCT_2:th.171">FUNCT_2:171</a></span>
;<br/><a NAME="E16:45_1_4_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c10">D</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">B</font> = <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c10">D</font> as   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></span>;<br/><a NAME="E18:45_1_4_2"/><span class="lab"><font color="Green" title="E53">A28</font></span>: 
<font color="Maroon" title="c10">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c11">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:45_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E2:45_1_4_2"><span class="lab"><font color="Green" title="E46">A21</font></span></a>, <a class="ref" href="funct_1.html#T147" title="FUNCT_1:th.147">FUNCT_1:147</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E19:45_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A29</font></span>: 
not <font color="Maroon" title="c11">B</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T149" title="RELAT_1:th.149">RELAT_1:149</a></span>;<br/><a NAME="E20:45_1_4_2"/><span class="lab"><font color="Green" title="E55">A30</font></span>: 
the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice2.html#T27" title="LATTICE2:th.27">LATTICE2:27</a></span>;<br/><a NAME="E21:45_1_4_2"/><span class="lab"><font color="Green" title="E56">A31</font></span>: 
the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice2.html#T29" title="LATTICE2:th.29">LATTICE2:29</a></span>;<br/><a NAME="E22:45_1_4_2"/><span class="lab"><font color="Green" title="E57">A32</font></span>: 
the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> is  <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice2.html#T26" title="LATTICE2:th.26">LATTICE2:26</a></span>;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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>,  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <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> = $1 <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> $2;<br/><span class="kw">consider </span><font color="Maroon" title="c12">G</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <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"> such that </span><br/><a NAME="E24:45_1_4_2"/><span class="lab"><font color="Green" title="E58">A33</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> holds  <font color="Maroon" title="c12">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></span><a NAME="E25:45_1_4_2"/><span class="lab"><font color="Green" title="E59">A34</font></span>: 
<font color="Maroon" title="c12">G</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_4_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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="comment"><font color="Red">:: according to </font></span><a class="ref" href="binop_1.html#D13" title="BINOP_1:def.13">BINOP_1:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c13">x</font></span><br/>

<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font> = 
<font color="Maroon" title="c14">y</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c13">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:45_1_4_2"><span class="lab"><font color="Green" title="E58">A33</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c13">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:45_1_4_2"><span class="lab"><font color="Green" title="E58">A33</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:45_1_4_2_2"/>
<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c13">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E26:45_1_4_2"/><span class="lab"><font color="Green" title="E60">A35</font></span>: 
<font color="Maroon" title="c12">G</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_4_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</font>, <font color="Maroon" title="c15">z</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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="comment"><font color="Red">:: according to </font></span><a class="ref" href="binop_1.html#D14" title="BINOP_1:def.14">BINOP_1:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c15">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font></span>)</span>,<font color="Maroon" title="c15">z</font></span><br/>

<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c13">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c15">z</font></span>)</span> = 
<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c13">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">y</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c15">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:45_1_4_2"><span class="lab"><font color="Green" title="E58">A33</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c13">x</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">y</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c15">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:45_1_4_2"><span class="lab"><font color="Green" title="E58">A33</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c15">z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></span>
<br/>.= 
<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c14">y</font></span>)</span>,<font color="Maroon" title="c15">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:45_1_4_2"><span class="lab"><font color="Green" title="E58">A33</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font></span>)</span>,<font color="Maroon" title="c15">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:45_1_4_2"><span class="lab"><font color="Green" title="E58">A33</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:45_1_4_2_3"/>
<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c15">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font></span>)</span>,<font color="Maroon" title="c15">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E27:45_1_4_2"/><span class="lab"><font color="Green" title="E61">A36</font></span>: 
<font color="Maroon" title="c12">G</font> is  <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_4_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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="comment"><font color="Red">:: according to </font></span><a class="ref" href="binop_1.html#D15" title="BINOP_1:def.15">BINOP_1:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font></span><br/>

<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c13">x</font> = 
<font color="Maroon" title="c13">x</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c13">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:45_1_4_2"><span class="lab"><font color="Green" title="E58">A33</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c13">x</font>

;<br/>
<span class="kw">hence </span><a NAME="E2:45_1_4_2_4"/>
<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E28:45_1_4_2"/><span class="lab"><font color="Green" title="E62">A37</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font> holds  <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_4_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font></span>)</span></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c14">y</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopclset.html#T24" target="_self" title="LOPCLSET:th.24">Th24</a></span>
<br/>.= 
<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:45_1_4_2"><span class="lab"><font color="Green" title="E58">A33</font></span></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/>


</div><span class="kw">end;</span></div><span class="kw">reconsider </span><font color="Maroon" title="c13">DD</font> = <font color="Maroon" title="c10">D</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</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="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></span>;<br/><a NAME="E30:45_1_4_2"/>
 <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font>
 ;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A38</font></span>: <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c11">B</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Maroon" title="c11">B</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice4.html#D12" title="LATTICE4:def.12">LATTICE4:def 12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c11">B</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice2.html#D3" title="LATTICE2:def.3">LATTICE2:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c11">B</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:45_1_4_2"><span class="lab"><font color="Green" title="E54">A29</font></span></a>, <a class="txt" href="#E20:45_1_4_2"><span class="lab"><font color="Green" title="E55">A30</font></span></a>, <a class="txt" href="#E21:45_1_4_2"><span class="lab"><font color="Green" title="E56">A31</font></span></a>, <a class="txt" href="#E22:45_1_4_2"><span class="lab"><font color="Green" title="E57">A32</font></span></a>, <a class="txt" href="#E25:45_1_4_2"><span class="lab"><font color="Green" title="E59">A34</font></span></a>, <a class="txt" href="#E26:45_1_4_2"><span class="lab"><font color="Green" title="E60">A35</font></span></a>, <a class="txt" href="#E27:45_1_4_2"><span class="lab"><font color="Green" title="E61">A36</font></span></a>, <a class="txt" href="#E28:45_1_4_2"><span class="lab"><font color="Green" title="E62">A37</font></span></a>, <a class="ref" href="setwiseo.html#T39" title="SETWISEO:th.39">SETWISEO:39</a></span>
<br/>.= 
<font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c11">B</font>,<span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></span>
<br/>.= 
<font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c11">B</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p4">(<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>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></span>
<br/>.= 
<font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c13">DD</font>,<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p3">(<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>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:45_1_4_2"><span class="lab"><font color="Green" title="E53">A28</font></span></a>, <a class="txt" href="#E19:45_1_4_2"><span class="lab"><font color="Green" title="E54">A29</font></span></a>, <a class="txt" href="#E25:45_1_4_2"><span class="lab"><font color="Green" title="E59">A34</font></span></a>, <a class="txt" href="#E26:45_1_4_2"><span class="lab"><font color="Green" title="E60">A35</font></span></a>, <a class="txt" href="#E27:45_1_4_2"><span class="lab"><font color="Green" title="E61">A36</font></span></a>, <a class="ref" href="setwiseo.html#T38" title="SETWISEO:th.38">SETWISEO:38</a></span>
;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</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">means </span>for <font color="Olive" title="b1">D</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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>  st <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> $1,<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p3">(<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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">D</font>;<br/><a NAME="E32:45_1_4_2"/><span class="lab"><font color="Green" title="E64">A39</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</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>]
 ;<br/><a NAME="E33:45_1_4_2"/><span class="lab"><font color="Green" title="E65">A40</font></span>: 
for <font color="Olive" title="b1">DD</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</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><br/> for <font color="Olive" title="b2">b</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">DD</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">DD</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_4_2_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">DD</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">b</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">DD</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">DD</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]</span><br/><span class="kw">let </span><font color="Maroon" title="c15">b</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">DD</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">DD</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c15">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>] )</span><br/>



<span class="kw">assume </span><a NAME="E1:45_1_4_2_7"/><span class="lab"><font color="Green" title="E66">A41</font></span>: 
for <font color="Olive" title="b1">D</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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>  st <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">DD</font> holds <br/><font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c14">DD</font>,<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p3">(<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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">DD</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c15">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c16">D</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">DD</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c15">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span> implies <font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">DD</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c15">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p3">(<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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c16">D</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:45_1_4_2_7"/><span class="lab"><font color="Green" title="E67">A42</font></span>: 
<font color="Maroon" title="c16">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">DD</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">DD</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c15">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p3">(<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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c16">D</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_4_2_7_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_4_2_7_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:45_1_4_2_7_1_1"/>
( <font color="Maroon" title="c14">DD</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c14">DD</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_4_2_7_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:45_1_4_2_7_1_1_1"/><span class="lab"><font color="Green" title="E68">A43</font></span>: 
<font color="Maroon" title="c14">DD</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">DD</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c15">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p3">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c16">D</font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">DD</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c15">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p3">(<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>)</span> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p3">(<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>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:45_1_4_2"><span class="lab"><font color="Green" title="E59">A34</font></span></a>, <a class="txt" href="#E26:45_1_4_2"><span class="lab"><font color="Green" title="E60">A35</font></span></a>, <a class="ref" href="setwiseo.html#T26" title="SETWISEO:th.26">SETWISEO:26</a></span>
<br/>.= 
<font color="Maroon" title="c15">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>
<br/>.= 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c16">D</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1_4_2_7"><span class="lab"><font color="Green" title="E67">A42</font></span></a>, <a class="txt" href="#E1:45_1_4_2_7_1_1_1"><span class="lab"><font color="Green" title="E68">A43</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_4_2_7_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:45_1_4_2_7_1_1_2"/><span class="lab"><font color="Green" title="E68">A44</font></span>: 
not <font color="Maroon" title="c14">DD</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">DD</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c15">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p3">(<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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c16">D</font></span><br/><div class="add"><a NAME="E2:45_1_4_2_7_1_1_2"/>
<font color="Maroon" title="c14">DD</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1_4_2_7"><span class="lab"><font color="Green" title="E67">A42</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">D1</font> = <font color="Maroon" title="c14">DD</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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">by</span> <span class="lab"><a class="txt" href="#E1:45_1_4_2_7_1_1_2"><span class="lab"><font color="Green" title="E68">A44</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">D1</font> = <font color="Maroon" title="c17">D1</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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> ;<br/><span class="kw">thus </span><font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">DD</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c15">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p3">(<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>)</span> = 
<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c14">DD</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p4">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p4">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:45_1_4_2"><span class="lab"><font color="Green" title="E59">A34</font></span></a>, <a class="txt" href="#E26:45_1_4_2"><span class="lab"><font color="Green" title="E60">A35</font></span></a>, <a class="txt" href="#E27:45_1_4_2"><span class="lab"><font color="Green" title="E61">A36</font></span></a>, <a class="txt" href="#E1:45_1_4_2_7_1_1_2"><span class="lab"><font color="Green" title="E68">A44</font></span></a>, <a class="ref" href="setwiseo.html#T29" title="SETWISEO:th.29">SETWISEO:29</a></span>
<br/>.= 
<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c14">DD</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p4">(<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>)</span></span>)</span>,<font color="Maroon" title="c15">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c14">DD</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p4">(<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>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c15">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:45_1_4_2"><span class="lab"><font color="Green" title="E58">A33</font></span></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="c18">D1</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c15">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_4_2_7"><span class="lab"><font color="Green" title="E66">A41</font></span></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="c18">D1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>
<br/>.= 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c16">D</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1_4_2_7"><span class="lab"><font color="Green" title="E67">A42</font></span></a>, <a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:45_1_4_2_7"/>
<font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">DD</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c15">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p3">(<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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c16">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E34:45_1_4_2"/>
for <font color="Olive" title="b1">DD</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</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> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">DD</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="setwiseo.html#S4" title="SETWISEO:sch.4">SETWISEO:sch 4</a>(<a class="txt" href="#E32:45_1_4_2"><span class="lab"><font color="Green" title="E64">A39</font></span></a>, <a class="txt" href="#E33:45_1_4_2"><span class="lab"><font color="Green" title="E65">A40</font></span></a>);<br/></span><a NAME="E35:45_1_4_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c11">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:45_1_4_2"><span class="lab"><font color="Green" title="E48">A23</font></span></a>, <a class="txt" href="#E31:45_1_4_2"><span class="lab"><font color="Green" title="E63">A38</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E36:45_1_4_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:45"/>
 <a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E28">A1</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/>


</div>
