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<b>let </b><font color="Maroon" title="c1">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K2" title="MSUALG_5:func.2">EqRelLatt</a> <font color="Maroon" title="c1">Y</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c3">R</font> be   <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of <font color="Maroon" title="c1">Y</font>;<br/>





<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font>
 ;<br/>

<a NAME="E2:11"/><b>then </b><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c3">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="msualg_8.html#T7" target="_self" title="MSUALG_8:th.7">Th7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">X1</font> =  <a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Maroon" title="c2">X</font> as   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">Y</font> <b>by </b><i><a class="ref" href="msualg_7.html#T1" title="MSUALG_7:th.1">MSUALG_7:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">EqR</font> be   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">Y</font>;<br/><b>reconsider </b><font color="Maroon" title="c6">EqR1</font> = <font color="Maroon" title="c5">EqR</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K2" title="MSUALG_5:func.2">EqRelLatt</a> <font color="Maroon" title="c1">Y</font></span>)</span> <b>by </b><i><a class="ref" href="msualg_7.html#T1" title="MSUALG_7:th.1">MSUALG_7:1</a></i>;<br/><b>assume </b><a NAME="E2:11_1"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c3">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">EqR</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K2" title="MSUALG_5:func.2">EqRelLatt</a> <font color="Maroon" title="c1">Y</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c8">q1</font> = <font color="Maroon" title="c7">q</font> as   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">Y</font> <b>by </b><i><a class="ref" href="msualg_7.html#T1" title="MSUALG_7:th.1">MSUALG_7:1</a></i>;<br/><b>assume </b><a NAME="E2:11_1_1"/><i><font color="Green" title="E10">A4</font></i>: 
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:11_1_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">q1</font>
 ;<br/><a NAME="E2:11_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="msualg_8.html#E2:11_1_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E3:11_1_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">EqR</font>
 <b>by </b><i><a class="txt" href="msualg_8.html#E1:11"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="msualg_8.html#E2:11_1"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E4:11_1_1"/><b>then </b>
<font color="Maroon" title="c8">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">EqR</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E5:11_1_1"/>
<font color="Maroon" title="c7">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">EqR1</font>
 <b>by </b><i><a class="ref" href="msualg_8.html#T2" target="_self" title="MSUALG_8:th.2">Th2</a></i>;<br/></div><b>end;</b></div><a NAME="E4:11_1"/><b>then </b>
<font color="Maroon" title="c2">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c6">EqR1</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a></i>;<br/><a NAME="E5:11_1"/><b>then </b>
 <a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Maroon" title="c2">X</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">EqR1</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>;<br/><b>hence </b><a NAME="E6:11_1"/>
<font color="Maroon" title="c4">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">EqR</font>
 <b>by </b><i><a class="ref" href="msualg_8.html#T2" target="_self" title="MSUALG_8:th.2">Th2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:11"/>
 <a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_5.html#K1" title="MSUALG_5:func.1">EqCl</a> <font color="Maroon" title="c3">R</font>
 <b>by </b><i><a class="txt" href="msualg_8.html#E2:11"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="msualg_5.html#D1" title="MSUALG_5:def.1">MSUALG_5:def 1</a></i>;<br/>


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