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<b>let </b><font color="Maroon">L1</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">L2</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>;<br/><b>let </b><font color="Maroon">q1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>;<br/><b>let </b><font color="Maroon">p2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font>;<br/><b>let </b><font color="Maroon">q2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font>;<br/>











<b>thus </b><a NAME="E1:56"/>
( <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#R3">[=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> implies ( <font color="Maroon">p1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">p2</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:56_1"/>
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K3">"\/"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D3">LATTICES:def 3</a><br/>

<a NAME="E2:56_1"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q2</font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:56_1"/>
( <font color="Maroon">p1</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">p2</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D3">LATTICES:def 3</a><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:56"/>
( <font color="Maroon">p1</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">p2</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D3">LATTICES:def 3</a><br/>

<b>hence </b><a NAME="E3:56"/>
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K3">"\/"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D3">LATTICES:def 3</a><br/>


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