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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">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">F</font> being   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c1">L</font> is   <a href="filter_0.html#NM2" title="FILTER_0:NM.2">I_Lattice</a> holds <br/> <a href="filter_0.html#K6" title="FILTER_0:func.6">latt</a> <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</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"> ( <font color="Maroon" title="c1">L</font> is   <a href="filter_0.html#NM2" title="FILTER_0:NM.2">I_Lattice</a> implies  <a href="filter_0.html#K6" title="FILTER_0:func.6">latt</a> <font color="Maroon" title="c2">F</font> is  <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:70"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
<font color="Maroon" title="c1">L</font> is   <a href="filter_0.html#NM2" title="FILTER_0:NM.2">I_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">  <a href="filter_0.html#K6" title="FILTER_0:func.6">latt</a> <font color="Maroon" title="c2">F</font> is  <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">I</font> = <font color="Maroon" title="c1">L</font> as   <a href="filter_0.html#NM2" title="FILTER_0:NM.2">I_Lattice</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">FI</font> = <font color="Maroon" title="c2">F</font> as   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c3">I</font> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">p9</font>, <font color="Maroon" title="c6">q9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="filter_0.html#K6" title="FILTER_0:func.6">latt</a> <font color="Maroon" title="c2">F</font></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="filter_0.html#D6" target="_self" onmouseover="rs('filter_0/D6')" onmouseout="rh()">FILTER_0:def 6</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">  ex <font color="Olive" title="b1">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="filter_0.html#K6" title="FILTER_0:func.6">latt</a> <font color="Maroon" title="c2">F</font></span>)</span> st <br/>( <font color="Maroon" title="c5">p9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b1">r</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">q9</font> &amp; (  for <font color="Olive" title="b2">r1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="filter_0.html#K6" title="FILTER_0:func.6">latt</a> <font color="Maroon" title="c2">F</font></span>)</span>  st <font color="Maroon" title="c5">p9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">r1</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">q9</font> holds <br/><font color="Olive" title="b2">r1</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive" title="b1">r</font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">p</font> = <font color="Maroon" title="c5">p9</font>, <font color="Maroon" title="c8">q</font> = <font color="Maroon" title="c6">q9</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">F</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T49" target="_self" onmouseover="rs('filter_0/T49')" onmouseout="rh()">Th49</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E6:70"/><span class="lab"><font color="Green" title="E48">A2</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">q</font>
 <span class="kw">and </span><br/><a NAME="E7:70"/><span class="lab"><font color="Green" title="E49">A3</font></span>: 
 for <font color="Olive" title="b1">r1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c7">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">q</font> holds <br/><font color="Olive" title="b1">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="ref" href="filter_0.html#D6" target="_self" onmouseover="rs('filter_0/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">i</font> = <font color="Maroon" title="c7">p</font>, <font color="Maroon" title="c11">j</font> = <font color="Maroon" title="c8">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">I</font> ;<br/>
<a NAME="E9:70"/><span class="lab"><font color="Green" title="E50">A4</font></span>: 
<font color="Maroon" title="c10">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c11">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">FI</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T30" target="_self" onmouseover="rs('filter_0/T30')" onmouseout="rh()">Th30</a></span>;<br/>
<a NAME="E10:70"/>
<font color="Maroon" title="c10">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="#E7:70"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="ref" href="filter_0.html#D7" target="_self" onmouseover="rs('filter_0/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">r9</font> = <font color="Maroon" title="c9">r</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="filter_0.html#K6" title="FILTER_0:func.6">latt</a> <font color="Maroon" title="c2">F</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:70"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="ref" href="filter_0.html#T49" target="_self" onmouseover="rs('filter_0/T49')" onmouseout="rh()">Th49</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c12">r9</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"> ( <font color="Maroon" title="c5">p9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c12">r9</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">q9</font> &amp; (  for <font color="Olive" title="b1">r1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="filter_0.html#K6" title="FILTER_0:func.6">latt</a> <font color="Maroon" title="c2">F</font></span>)</span>  st <font color="Maroon" title="c5">p9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b1">r1</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">q9</font> holds <br/><font color="Olive" title="b1">r1</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c12">r9</font> ) )</span><br/>

<a NAME="E12:70"/>
<font color="Maroon" title="c7">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c12">r9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T50" target="_self" onmouseover="rs('filter_0/T50')" onmouseout="rh()">Th50</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:70"/>
<font color="Maroon" title="c5">p9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c12">r9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="filter_0.html#T51" target="_self" onmouseover="rs('filter_0/T51')" onmouseout="rh()">Th51</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">  for <font color="Olive" title="b1">r1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="filter_0.html#K6" title="FILTER_0:func.6">latt</a> <font color="Maroon" title="c2">F</font></span>)</span>  st <font color="Maroon" title="c5">p9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b1">r1</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">q9</font> holds <br/><font color="Olive" title="b1">r1</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c12">r9</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c13">s9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="filter_0.html#K6" title="FILTER_0:func.6">latt</a> <font color="Maroon" title="c2">F</font></span>)</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"> ( <font color="Maroon" title="c5">p9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c13">s9</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">q9</font> implies <font color="Maroon" title="c13">s9</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c12">r9</font> )</span><br/>

<span class="kw">assume </span><a NAME="E14:70"/><span class="lab"><font color="Green" title="E51">A5</font></span>: 
<font color="Maroon" title="c5">p9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c13">s9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">q9</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"> <font color="Maroon" title="c13">s9</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c12">r9</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">s</font> = <font color="Maroon" title="c13">s9</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">F</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T49" target="_self" onmouseover="rs('filter_0/T49')" onmouseout="rh()">Th49</a></span>;<br/>
<a NAME="E16:70"/>
<font color="Maroon" title="c7">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c13">s9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T50" target="_self" onmouseover="rs('filter_0/T50')" onmouseout="rh()">Th50</a></span>;<br/>
<a NAME="E17:70"/><span class="kw">then </span>
<font color="Maroon" title="c7">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c14">s</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:70"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="ref" href="filter_0.html#T51" target="_self" onmouseover="rs('filter_0/T51')" onmouseout="rh()">Th51</a></span>;<br/>
<a NAME="E18:70"/><span class="kw">then </span>
<font color="Maroon" title="c14">s</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70"><span class="lab"><font color="Green" title="E49">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:70"/>
<font color="Maroon" title="c13">s9</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c12">r9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T51" target="_self" onmouseover="rs('filter_0/T51')" onmouseout="rh()">Th51</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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