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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:56"/><i><font color="Green">E22</font></i>: 
(  <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>2</sub></font> #) &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E2:56"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K3" target="_self">.)</a></span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="filter_2.html#M1" target="_self">Filter</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K3" target="_self">.)</a></span> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="filter_2.html#K3" target="_self">.)</a></span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="filter_2.html#M1" target="_self">Filter</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="filter_2.html#K3" target="_self">.)</a></span> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K3" target="_self">.)</a></span> is    <a href="filter_2.html#M1" target="_self">Filter</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="filter_2.html#K3" target="_self">.)</a></span> is    <a href="filter_2.html#M1" target="_self">Filter</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="filter_2.html#T16" target="_self">Th16</a>, <a class="ref" href="filter_0.html#D5">FILTER_0:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:56"/>
( <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K3" target="_self">.)</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="filter_2.html#K3" target="_self">.)</a></span> &amp; <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="filter_2.html#K3" target="_self">.)</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K3" target="_self">.)</a></span> )
 <b>by </b><i><a class="txt" href="filter_2.html#E1:56"><i><font color="Green">E22</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<a NAME="E4:56"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="filter_2.html#M2" target="_self">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="filter_2.html#M2" target="_self">Ideal</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> is    <a href="filter_2.html#M2" target="_self">Ideal</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> is    <a href="filter_2.html#M2" target="_self">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="filter_2.html#E1:56"><i><font color="Green">E22</font></i></a>, <a class="ref" href="filter_2.html#D11" target="_self">Def11</a>, <a class="ref" href="filter_2.html#T17" target="_self">Th17</a></i>;<br/>
<b>hence </b><a NAME="E5:56"/>
( <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> &amp; <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> )
 <b>by </b><i><a class="txt" href="filter_2.html#E1:56"><i><font color="Green">E22</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>


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