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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:88"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is   <a href="filter_0.html#NM2">I_Lattice</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="filter_0.html#K10" target="_self">equivalence_wrt</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="relset_1.html#NM3">Relation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="filter_0.html#T80" target="_self">Th80</a></i>;<br/>
<a NAME="E3:88"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="partfun1.html#V1">total</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="relat_2.html#V3">symmetric</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="relat_2.html#V8">transitive</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><b>proof </b></a><div class="add">

<a NAME="E1:88_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="filter_0.html#E1:88"><i><font color="Green">E63</font></i></a>, <a class="ref" href="filter_0.html#T81" target="_self">Th81</a></i>;<br/>
<a NAME="E2:88_1"/><b>then </b><i><font color="Green">E64</font></i>: 
(  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="orders_1.html#T98">ORDERS_1:98</a></i>;<br/>
<b>hence </b><a NAME="E3:88_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="partfun1.html#V1">total</a>
 <b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>

<a NAME="E4:88_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="relat_2.html#R3">is_symmetric_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="relat_2.html#R8">is_transitive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="filter_0.html#E1:88"><i><font color="Green">E63</font></i></a>, <a class="ref" href="filter_0.html#T82" target="_self">Th82</a>, <a class="ref" href="filter_0.html#T83" target="_self">Th83</a></i>;<br/>
<b>hence </b><a NAME="E5:88_1"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="relat_2.html#V3">symmetric</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="relat_2.html#V8">transitive</a> )
 <b>by </b><i><a class="txt" href="filter_0.html#E2:88_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="relat_2.html#D11">RELAT_2:def 11</a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:88"/>
 <a href="filter_0.html#K10" target="_self">equivalence_wrt</a> <font color="Maroon">c<sub>2</sub></font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


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