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<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>, <font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:51_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="filter_0.html#K3" target="_self">.)</a></span>
 <b>by </b><i><a class="ref" href="filter_0.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E2:51_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="filter_0.html#K3" target="_self">.)</a></span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/><b>hence </b><a NAME="E3:51_1"/>
<span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="filter_0.html#K3" target="_self">.)</a></span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span>
 <b>by </b><i><a class="ref" href="filter_0.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E4:51_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> )
 <b>by </b><i><a class="ref" href="filter_0.html#D5" target="_self">Def5</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E5:51_1"/><b>then </b>
( <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="filter_0.html#K3" target="_self">.)</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> )
 <b>by </b><i><a class="ref" href="filter_0.html#T27" target="_self">Th27</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E6:51_1"/><b>then </b>
<span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="filter_0.html#K3" target="_self">.)</a></span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E7:51_1"/>
<span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="filter_0.html#K3" target="_self">.)</a></span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span>
 <b>by </b><i><a class="ref" href="filter_0.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:51"/>
( <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="filter_0.html#K3" target="_self">.)</a></span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> &amp; <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="filter_0.html#K3" target="_self">.)</a></span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> &amp; <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="filter_0.html#K3" target="_self">.)</a></span></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> &amp; <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="filter_0.html#K3" target="_self">.)</a></span></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>


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