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<div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><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="E1:26"/><i><font color="Green">E25</font></i>: 
( <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> implies  ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">BL</font> st <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> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_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/>

<a NAME="E2:26_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>
 
;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:26_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</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; <font color="Maroon">x</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="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">BL</font> <b>by </b><i><a class="txt" href="lopclset.html#E4:26_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>thus </b><a NAME="E6:26_1"/>
<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> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="lopclset.html#E4:26_1"><i><font color="Green">E26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:26"/>
(  ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">BL</font> st <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> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> implies <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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">BL</font><b> such that </b><a NAME="E1:26_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</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="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:26_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font>
 
;<br/>
<a NAME="E3:26_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</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>
 
<b>by </b><i/>;<br/>
<a NAME="E4:26_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_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:26_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>
<b>hence </b><a NAME="E3:26"/>
( <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> iff  ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">BL</font> st <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> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="lopclset.html#E1:26"><i><font color="Green">E25</font></i></a></i>;<br/>


</div>
