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<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K2">EqRelLatt</a> <font color="Maroon">Y</font></span>)</span>;<br/><b>let </b><font color="Maroon">R</font> be   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">Y</font>;<br/>





<b>assume </b><a NAME="E1:11"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:11"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">R</font> <a href="tarski.html#R1">c=</a>  <a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">X1</font> =  <a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">X</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="msualg_7.html#T1">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">EqR</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">Y</font>;<br/><b>reconsider </b><font color="Maroon">EqR1</font> = <font color="Maroon">EqR</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K2">EqRelLatt</a> <font color="Maroon">Y</font></span>)</span> <b>by </b><i><a class="ref" href="msualg_7.html#T1">MSUALG_7:1</a></i>;<br/><b>assume </b><a NAME="E2:11_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">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">q</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K2">EqRelLatt</a> <font color="Maroon">Y</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">q</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="msualg_7.html#T1">MSUALG_7:1</a></i>;<br/><b>assume </b><a NAME="E2:11_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:11_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">q1</font>
 ;<br/><a NAME="E2:11_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E3:11_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E4:11_1_1"/><b>then </b>
<font color="Maroon">q1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E5:11_1_1"/>
<font color="Maroon">q</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">EqR1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E4:11_1"/><b>then </b>
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">EqR1</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/><a NAME="E5:11_1"/><b>then </b>
 <a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">X</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">EqR1</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/><b>hence </b><a NAME="E6:11_1"/>
<font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:11"/>
 <a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="msualg_5.html#K1">EqCl</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="msualg_5.html#D1">MSUALG_5:def 1</a></i>;<br/>


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