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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM2" title="LATTICES:NM.2">D_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">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st  not <font color="Olive" title="b2">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">a</font> holds <br/><span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Olive" title="b2">b</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" title="OPENLATT:func.10">SF_have</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" title="OPENLATT:func.10">SF_have</a> <font color="Olive" title="b1">a</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"> (  not <font color="Maroon" title="c3">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> implies <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" title="OPENLATT:func.10">SF_have</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" title="OPENLATT:func.10">SF_have</a> <font color="Maroon" title="c2">a</font></span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:32"/>
 not <font color="Maroon" title="c3">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</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"> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" title="OPENLATT:func.10">SF_have</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" title="OPENLATT:func.10">SF_have</a> <font color="Maroon" title="c2">a</font></span>)</span></span><br/>

<a NAME="E2:32"/><span class="kw">then </span>
 not <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T15" onmouseover="rs('filter_0/T15')" onmouseout="rh()">FILTER_0:15</a></span>;<br/>
<a NAME="E3:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A1</font></span>: 
 not <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="openlatt.html#K10" title="OPENLATT:func.10">SF_have</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="openlatt.html#T16" target="_self" onmouseover="rs('openlatt/T16')" onmouseout="rh()">Th16</a></span>;<br/>
<a NAME="E4:32"/>
<font color="Maroon" title="c3">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span>
 
;<br/>
<a NAME="E5:32"/><span class="kw">then </span>
<span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="openlatt.html#K10" title="OPENLATT:func.10">SF_have</a> <font color="Maroon" title="c3">b</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E6:32"/>
<span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" title="OPENLATT:func.10">SF_have</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" title="OPENLATT:func.10">SF_have</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</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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