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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="waybel_0.html#NM6" title="WAYBEL_0:NM.6">Semilattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font><br/>  for <font color="Olive" title="b2">G</font> being    <a href="waybel12.html#M1" title="WAYBEL12:mode.1">GeneratorSet</a> of <font color="Olive" title="b1">F</font><br/>  for <font color="Olive" title="b3">A</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">G</font> <a href="yellow_4.html#R4" title="YELLOW_4:pred.4">is_coarser_than</a> <font color="Olive" title="b3">A</font> &amp; <font color="Olive" title="b3">A</font> <a href="yellow_4.html#R4" title="YELLOW_4:pred.4">is_coarser_than</a> <font color="Olive" title="b1">F</font> holds <br/><font color="Olive" title="b3">A</font> is    <a href="waybel12.html#M1" title="WAYBEL12:mode.1">GeneratorSet</a> of <font color="Olive" title="b1">F</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G</font> being    <a href="waybel12.html#M1" title="WAYBEL12:mode.1">GeneratorSet</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b2">A</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">G</font> <a href="yellow_4.html#R4" title="YELLOW_4:pred.4">is_coarser_than</a> <font color="Olive" title="b2">A</font> &amp; <font color="Olive" title="b2">A</font> <a href="yellow_4.html#R4" title="YELLOW_4:pred.4">is_coarser_than</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Olive" title="b2">A</font> is    <a href="waybel12.html#M1" title="WAYBEL12:mode.1">GeneratorSet</a> of <font color="Maroon" title="c2">F</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">G</font> be    <a href="waybel12.html#M1" title="WAYBEL12:mode.1">GeneratorSet</a> of <font color="Maroon" title="c2">F</font>; <a class="txt" 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> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c3">G</font> <a href="yellow_4.html#R4" title="YELLOW_4:pred.4">is_coarser_than</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b1">A</font> <a href="yellow_4.html#R4" title="YELLOW_4:pred.4">is_coarser_than</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Olive" title="b1">A</font> is    <a href="waybel12.html#M1" title="WAYBEL12:mode.1">GeneratorSet</a> of <font color="Maroon" title="c2">F</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">A</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">G</font> <a href="yellow_4.html#R4" title="YELLOW_4:pred.4">is_coarser_than</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c4">A</font> <a href="yellow_4.html#R4" title="YELLOW_4:pred.4">is_coarser_than</a> <font color="Maroon" title="c2">F</font> implies <font color="Maroon" title="c4">A</font> is    <a href="waybel12.html#M1" title="WAYBEL12:mode.1">GeneratorSet</a> of <font color="Maroon" title="c2">F</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:41"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c3">G</font> <a href="yellow_4.html#R4" title="YELLOW_4:pred.4">is_coarser_than</a> <font color="Maroon" title="c4">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">A</font> <a href="yellow_4.html#R4" title="YELLOW_4:pred.4">is_coarser_than</a> <font color="Maroon" title="c2">F</font> or <font color="Maroon" title="c4">A</font> is    <a href="waybel12.html#M1" title="WAYBEL12:mode.1">GeneratorSet</a> of <font color="Maroon" title="c2">F</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:41"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="yellow_4.html#R4" title="YELLOW_4:pred.4">is_coarser_than</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">A</font> is    <a href="waybel12.html#M1" title="WAYBEL12:mode.1">GeneratorSet</a> of <font color="Maroon" title="c2">F</font></span><br/>

<a NAME="E3:41"/>
<font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c3">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel12.html#D3" target="_self" title="WAYBEL12:def.3">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:41"/>
<font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c4">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="waybel12.html#T4" target="_self" title="WAYBEL12:th.4">Th8</a>, <a class="ref" href="waybel12.html#T29" target="_self" title="WAYBEL12:th.29">Th33</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>,<a class="ref" href="waybel12.html#D3" target="_self" title="WAYBEL12:def.3">WAYBEL12:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font></span><br/>

<a NAME="E5:41"/>
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="yellow_4.html#T8" title="YELLOW_4:th.8">YELLOW_4:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:41"/>
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T62" title="WAYBEL_0:th.62">WAYBEL_0:62</a></span>; <a class="txt" 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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