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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c1">L</font> is   <a href="filter_0.html#NM2" title="FILTER_0:NM.2">I_Lattice</a> holds <br/> <a href="filter_0.html#K8" title="FILTER_0:func.8">equivalence_wrt</a> <font color="Olive" title="b1">F</font> is   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is   <a href="filter_0.html#NM2" title="FILTER_0:NM.2">I_Lattice</a> implies  <a href="filter_0.html#K8" title="FILTER_0:func.8">equivalence_wrt</a> <font color="Maroon" title="c2">F</font> is   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c3">R</font> =  <a href="filter_0.html#K8" title="FILTER_0:func.8">equivalence_wrt</a> <font color="Maroon" title="c2">F</font> as   <a href="relset_1.html#NM2" title="RELSET_1:NM.2">Relation</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T63" target="_self" onmouseover="rs('filter_0/T63')" onmouseout="rh()">Th63</a></span>;<br/>
<a NAME="E2:83"/><span class="lab"><font color="Green" title="E58">A1</font></span>: 
<font color="Maroon" title="c3">R</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T65" target="_self" onmouseover="rs('filter_0/T65')" onmouseout="rh()">Th65</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:83"/><span class="lab"><font color="Green" title="E59">A2</font></span>: 
<font color="Maroon" title="c1">L</font> is   <a href="filter_0.html#NM2" title="FILTER_0:NM.2">I_Lattice</a>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="filter_0.html#K8" title="FILTER_0:func.8">equivalence_wrt</a> <font color="Maroon" title="c2">F</font> is   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E4:83"/><span class="kw">then </span>
<font color="Maroon" title="c3">R</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T64" target="_self" onmouseover="rs('filter_0/T64')" onmouseout="rh()">Th64</a></span>;<br/>
<a NAME="E5:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A3</font></span>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <font color="Maroon" title="c3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#T13" onmouseover="rs('orders_1/T13')" onmouseout="rh()">ORDERS_1:13</a></span>;<br/>
<a NAME="E6:83"/>
<font color="Maroon" title="c3">R</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="filter_0.html#T66" target="_self" onmouseover="rs('filter_0/T66')" onmouseout="rh()">Th66</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:83"/>
 <a href="filter_0.html#K8" title="FILTER_0:func.8">equivalence_wrt</a> <font color="Maroon" title="c2">F</font> is   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a>, <a class="ref" href="relat_2.html#D11" onmouseover="rs('relat_2/D11')" onmouseout="rh()">RELAT_2:def 11</a>, <a class="ref" href="relat_2.html#D16" onmouseover="rs('relat_2/D16')" onmouseout="rh()">RELAT_2:def 16</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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