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

<b>let </b><font color="Maroon">BL</font> be  non <a href="realset2.html#V3">trivial</a> <a href="lattices.html#NM8">B_Lattice</a>;<br/>

<a NAME="E1:42"/><i><font color="Green">E25</font></i>: 
 <a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">BL</font> <a href="tarski.html#R1">c=</a>  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <span class="p1">(<span class="default"><a href="lopclset.html#K11" target="_self">StoneSpace</a> <font color="Maroon">BL</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:42"/>
 <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <span class="p1">(<span class="default"><a href="lopclset.html#K11" target="_self">StoneSpace</a> <font color="Maroon">BL</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">BL</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:42_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <span class="p1">(<span class="default"><a href="lopclset.html#K11" target="_self">StoneSpace</a> <font color="Maroon">BL</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K11" target="_self">StoneSpace</a> <font color="Maroon">BL</font></span>)</span> ;<br/>
<a NAME="E3:42_1"/><i><font color="Green">E46</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K11" target="_self">StoneSpace</a> <font color="Maroon">BL</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span> : <font color="Olive">A</font> <a href="tarski.html#R1">c=</a>  <a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">BL</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="lopclset.html#E1:42"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E4:42_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">X</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">X</font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="ref" href="lopclset.html#T7" target="_self">Th7</a>, , </i>;<br/>
<a NAME="E5:42_1"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a>, <a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/>
<a NAME="E6:42_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K11" target="_self">StoneSpace</a> <font color="Maroon">BL</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lopclset.html#T9" target="_self">Th9</a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">A1</font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span><b> such that </b><br/><a NAME="E8:42_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a>  <a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">BL</font> )
 <b>by </b><i><a class="ref" href="lopclset.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E9:42_1"/>
<font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K11" target="_self">StoneSpace</a> <font color="Maroon">BL</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lopclset.html#T10" target="_self">Th10</a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">A2</font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span><b> such that </b><br/><a NAME="E11:42_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a>  <a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">BL</font> )
 <b>by </b><i><a class="ref" href="lopclset.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E12:42_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">X</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lopclset.html#E1:42"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">AA</font> = <font color="Maroon">A1</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">A2</font>;<br/>
<i><font color="Green">E56</font></i>:  <a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="lopclset.html#K11" target="_self">StoneSpace</a> <font color="Maroon">BL</font></span>)</span>
<b>by </b><i><a class="txt" href="lopclset.html#E1:42"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<font color="Maroon">X</font> <a href="finsub_1.html#K1">\/</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T18">PRE_TOPC:18</a></i>
<br/>.= 
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">A2</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T96">ZFMISC_1:96</a></i>
;<br/>
<a NAME="E14:42_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">A1</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a>  <a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">BL</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being   <a href="finsub_1.html#NM1">Finite_Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">A2</font></span>)</span><b> such that </b><br/><a NAME="E16:42_1"/><i><font color="Green">E58</font></i>: 
 <a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="lopclset.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E17:42_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">A2</font>
 
<b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>
<a NAME="E18:42_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">BL</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>set </b><font color="Maroon">D</font> = <font color="Maroon">A1</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">Y</font>;<br/>
<i><font color="Green">E103</font></i>: <font color="Maroon">Y</font> = 
<span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">Y</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">Y</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:42_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">Y</font>
 ;<br/><a NAME="E2:42_1_3"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:42_1_3"/><b>then </b>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E4:42_1_3"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">y</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E5:42_1_3"/>
<font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><a NAME="E6:42_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E7:42_1_3"/><b>then </b>
<font color="Maroon">y</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E8:42_1_3"/>
<font color="Maroon">y</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E21:42_1"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="lopclset.html#E3:42_1"><i><font color="Green">E46</font></i></a>, , <a class="ref" href="setfam_1.html#T57">SETFAM_1:57</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_4"/>
( <font color="Maroon">A1</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">A1</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_4_1"><b>suppose </b></a><a NAME="E1:42_1_4_1"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:42_1_4_1"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/><a NAME="E3:42_1_4_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">BL</font></span>)</span>
 <b>by </b><i><a class="ref" href="lopclset.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E4:42_1_4_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span>
 <b>by </b><i><a class="txt" href="lopclset.html#E4:42_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/><b>hence </b><a NAME="E5:42_1_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">BL</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_4_2"><b>suppose </b></a><a NAME="E1:42_1_4_2"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">A1</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:42_1_4_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">A1</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">A1</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">Y</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span> <b>by </b><i><a class="txt" href="lopclset.html#E4:42_1"><i><font color="Green">E51</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">D</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span> ;<br/><a NAME="E5:42_1_4_2"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:42_1_4_2_1"/><i><font color="Green">E112</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T97">ZFMISC_1:97</a></i>;<br/>
<a NAME="E2:42_1_4_2_1"/>
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">Y</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">A1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E3:42_1_4_2_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="txt" href="lopclset.html#E8:42_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:42_1_4_2"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E7:42_1_4_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="lopclset.html#E5:42_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:42_1_4_2"/><i><font color="Green">E115</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">BL</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E9:42_1_4_2"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> &amp;  <a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">BL</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> )
 <b>by </b><i><a class="ref" href="lopclset.html#D2" target="_self">Def2</a>, <a class="ref" href="relat_1.html#T80">RELAT_1:80</a></i>;<br/><a NAME="E10:42_1_4_2"/><b>then </b>
 <a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font>, <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">BL</font></span>)</span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="txt" href="lopclset.html#E12:42_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="rfinseq.html#T3">RFINSEQ:3</a></i>;<br/><b>then </b> <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">D</font></span>)</span> = 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3">(<span class="default"><a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">BL</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">D</font></span>)</span>
<b>by </b><i><a class="ref" href="rfinseq.html#T4">RFINSEQ:4</a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">D</font>
<b>by </b><i><a class="txt" href="lopclset.html#E11:42_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="borsuk_1.html#T4">BORSUK_1:4</a></i>
;<br/><a NAME="E12:42_1_4_2"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">D</font></span>)</span> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="card_4.html#T1">CARD_4:1</a></i>;<br/><a NAME="E13:42_1_4_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">D</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="card_4.html#T1">CARD_4:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">B</font> = <span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">D</font> as   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font> <b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/><a NAME="E15:42_1_4_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="lopclset.html#E11:42_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#T147">FUNCT_1:147</a></i>;<br/><a NAME="E16:42_1_4_2"/><b>then </b><i><font color="Green">E117</font></i>: 
not <font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="relat_1.html#T149">RELAT_1:149</a></i>;<br/><a NAME="E17:42_1_4_2"/><i><font color="Green">E118</font></i>: 
( the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">BL</font> is <a href="binop_1.html#V1">commutative</a> &amp; the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">BL</font> is <a href="binop_1.html#V2">associative</a> &amp; the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">BL</font> is <a href="binop_1.html#V3">idempotent</a> )
 <b>by </b><i><a class="ref" href="lattice2.html#T26">LATTICE2:26</a>, <a class="ref" href="lattice2.html#T27">LATTICE2:27</a>, <a class="ref" href="lattice2.html#T29">LATTICE2:29</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span>,  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span> = <font color="Maroon">a<sub>1</sub></font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">a<sub>2</sub></font>;<br/><b>consider </b><font color="Maroon">G</font> being   <a href="binop_1.html#NM2">BinOp</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span><b> such that </b><br/><a NAME="E19:42_1_4_2"/><i><font color="Green">E119</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span> holds  <font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4">BINOP_1:sch 4</a>();<br/></i><a NAME="E20:42_1_4_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">G</font> is <a href="binop_1.html#V1">commutative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span>, <font color="Maroon">y</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D13">BINOP_1:def 13</a><br/>

<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<font color="Maroon">y</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="lopclset.html#T11" target="_self">Th11</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="lopclset.html#T11" target="_self">Th11</a></i>
;<br/>
<b>hence </b><a NAME="E2:42_1_4_2_3"/>
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E21:42_1_4_2"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">G</font> is <a href="binop_1.html#V2">associative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span>, <font color="Maroon">y</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span>, <font color="Maroon">z</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D14">BINOP_1:def 14</a><br/>

<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lopclset.html#T11" target="_self">Th11</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="finsub_1.html#K1">\/</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lopclset.html#T11" target="_self">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">y</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="lopclset.html#T11" target="_self">Th11</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="lopclset.html#T11" target="_self">Th11</a></i>
;<br/>
<b>hence </b><a NAME="E2:42_1_4_2_4"/>
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E22:42_1_4_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">G</font> is <a href="binop_1.html#V3">idempotent</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D15">BINOP_1:def 15</a><br/>

<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">x</font> = 
<font color="Maroon">x</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="lopclset.html#T11" target="_self">Th11</a></i>
<br/>.= 
<font color="Maroon">x</font>

;<br/>
<b>hence </b><a NAME="E2:42_1_4_2_5"/>
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E23:42_1_4_2"/><i><font color="Green">E123</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">BL</font> holds  <span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">BL</font> <a href="binop_1.html#K2">.</a> <font color="Olive">x</font>,<font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">BL</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">BL</font>;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">BL</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="lopclset.html#T11" target="_self">Th11</a></i>
;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">DD</font> = <font color="Maroon">D</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/><a NAME="E25:42_1_4_2"/>
 <a href="grcat_1.html#K7">id</a> <font color="Maroon">BL</font> <a href="funct_2.html#R2">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font>
 <b>by </b><i><a class="ref" href="grcat_1.html#D11">GRCAT_1:def 11</a></i>;<br/><b>then </b><i><font color="Green">E124</font></i>: <span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1">FinJoin</a> <font color="Maroon">B</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Maroon">B</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#D12">LATTICE4:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">BL</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice2.html#D3">LATTICE2:def 3</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lopclset.html#E14:42_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="lopclset.html#E16:42_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="lopclset.html#T12" target="_self">Th12</a>, <a class="ref" href="lopclset.html#T14" target="_self">Th14</a>, , , <a class="ref" href="setwiseo.html#T39">SETWISEO:39</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p4">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">DD</font>,<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p3">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lopclset.html#E13:42_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="lopclset.html#E14:42_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="lopclset.html#T12" target="_self">Th12</a>, <a class="ref" href="lopclset.html#T14" target="_self">Th14</a>, , <a class="ref" href="setwiseo.html#T38">SETWISEO:38</a></i>
;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span></span>)</span>] <b>means </b>for <font color="Olive">D</font> being non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span>  st <font color="Olive">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">G</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">a<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p3">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Olive">D</font>;<br/><a NAME="E27:42_1_4_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1">{}.</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span></span>)</span>]
 ;<br/><a NAME="E28:42_1_4_2"/><i><font color="Green">E126</font></i>: 
for <font color="Olive">DD</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span></span>)</span><br/> for <font color="Olive">b</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">DD</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">DD</font> <a href="setwiseo.html#K5">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Olive">b</font></span><a href="setwiseo.html#K2">.}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">DD</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:42_1_4_2_8"/><i><font color="Green">E127</font></i>: 
for <font color="Olive">D</font> being non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span>  st <font color="Olive">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">DD</font> holds <br/><font color="Maroon">G</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">DD</font>,<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p3">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Olive">D</font>
 ;<br/>

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span>;<br/>

<b>assume </b><a NAME="E2:42_1_4_2_8"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">DD</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4_2_8_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_4_2_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_4_2_8_1_1"/>
( <font color="Maroon">DD</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">DD</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_4_2_8_1_1_1"><b>suppose </b></a><a NAME="E1:42_1_4_2_8_1_1_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">DD</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">G</font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><font color="Maroon">DD</font> <a href="setwiseo.html#K5">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">b</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p3">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p3">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lopclset.html#T12" target="_self">Th12</a>, <a class="ref" href="lopclset.html#T14" target="_self">Th14</a>, <a class="ref" href="setwiseo.html#T26">SETWISEO:26</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
<br/>.= 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font>
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_4_2_8_1_1_2"><b>suppose </b></a><a NAME="E1:42_1_4_2_8_1_1_2"/><i><font color="Green">E146</font></i>: 
not <font color="Maroon">DD</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:42_1_4_2_8_1_1_2"/>
<font color="Maroon">DD</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>then reconsider </b><font color="Maroon">D1</font> = <font color="Maroon">DD</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span> <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>reconsider </b><font color="Maroon">D1</font> = <font color="Maroon">D1</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span> ;<br/><b>thus </b><font color="Maroon">G</font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><font color="Maroon">DD</font> <a href="setwiseo.html#K5">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">b</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p3">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span></span>)</span></span>)</span> = 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">DD</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p4">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p4">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lopclset.html#T12" target="_self">Th12</a>, <a class="ref" href="lopclset.html#T14" target="_self">Th14</a>, , , <a class="ref" href="setwiseo.html#T29">SETWISEO:29</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">DD</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p4">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">DD</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p4">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lopclset.html#T11" target="_self">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">D1</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">D1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>
<br/>.= 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font>
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T96">ZFMISC_1:96</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:42_1_4_2_8"/>
<font color="Maroon">G</font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><font color="Maroon">DD</font> <a href="setwiseo.html#K5">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">b</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p3">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E29:42_1_4_2"/>
for <font color="Olive">DD</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">DD</font>]
 <b>from </b><i><a class="ref" href="setwiseo.html#S4">SETWISEO:sch 4</a>(, );<br/></i><a NAME="E30:42_1_4_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1">FinJoin</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="lopclset.html#E8:42_1"><i><font color="Green">E53</font></i></a>, </i>;<br/><a NAME="E31:42_1_4_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/><b>hence </b><a NAME="E32:42_1_4_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">BL</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:42"/>
 <a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">BL</font> <a href="hidden.html#R1">=</a>  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <span class="p1">(<span class="default"><a href="lopclset.html#K11" target="_self">StoneSpace</a> <font color="Maroon">BL</font></span>)</span>
 <b>by </b><i><a class="ref" href="lopclset.html#T6" target="_self">Th6</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
