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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="realset2.html#V3">trivial</a> <a href="lattice2.html#NM2">H_Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<a NAME="E1:56"/><i><font color="Green">E46</font></i>: 
 <a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font> is <a href="lattice4.html#V1">monomorphism</a>
 
<b>by </b><i><a class="ref" href="openlatt.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E2:56"/><i><font color="Green">E47</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K6" target="_self">Open_setLatt</a> <span class="p2">(<span class="default"><a href="openlatt.html#K17" target="_self">HTopSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="openlatt.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E3:56"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="filter_0.html#D8">FILTER_0:def 8</a></i>;<br/>
<a NAME="E4:56"/><b>then </b>
<span class="p1">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="lattice4.html#T7">LATTICE4:7</a></i>;<br/>
<a NAME="E5:56"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="lattice4.html#D1">LATTICE4:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K6" target="_self">Open_setLatt</a> <span class="p2">(<span class="default"><a href="openlatt.html#K17" target="_self">HTopSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/><b>assume </b><a NAME="E1:56_1"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:56_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K6" target="_self">Open_setLatt</a> <span class="p2">(<span class="default"><a href="openlatt.html#K17" target="_self">HTopSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:56_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="openlatt.html#E2:56"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E5:56_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="openlatt.html#E1:56_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="openlatt.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E6:56_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="openlatt.html#E4:56_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="openlatt.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E7:56_1"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3">INTERSECTION</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T36">SETFAM_1:36</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:56_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:56_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="openlatt.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E4:56_1_1"/>
<span class="p1">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:56_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K3">INTERSECTION</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="openlatt.html#E1:56_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="setfam_1.html#D5">SETFAM_1:def 5</a></i>;<br/><a NAME="E6:56_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="openlatt.html#E7:56_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="openlatt.html#E3:56_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="setfam_1.html#T56">SETFAM_1:56</a></i>;<br/><a NAME="E7:56_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="openlatt.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E8:56_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="openlatt.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E9:56_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="openlatt.html#E1:56"><i><font color="Green">E46</font></i></a>, <a class="ref" href="lattice4.html#T8">LATTICE4:8</a></i>;<br/><a NAME="E10:56_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="filter_0.html#D8">FILTER_0:def 8</a></i>;<br/><a NAME="E11:56_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice4.html#T7">LATTICE4:7</a></i>;<br/><b>hence </b><a NAME="E12:56_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="openlatt.html#E3:56_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="openlatt.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div><a NAME="E9:56_1"/><b>then </b>
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T94">ZFMISC_1:94</a></i>;<br/><b>hence </b><a NAME="E10:56_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="openlatt.html#E4:56_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="openlatt.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:56"/>
<span class="p1">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="filter_0.html#K4">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="openlatt.html#K18" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="openlatt.html#E5:56"><i><font color="Green">E48</font></i></a>, <a class="ref" href="filter_0.html#D8">FILTER_0:def 8</a></i>;<br/>


</div>
