<?xml version="1.0"?>
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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">p</font>, <font color="Olive" title="b2">q</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="c1">L</font> is  <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a>  &amp; <font color="Olive" title="b1">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">q</font> holds <br/> <a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K9" title="FILTER_2:func.9">[#</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><a href="filter_2.html#K9" title="FILTER_2:func.9">#]</a></span>) 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">p</font>, <font color="Maroon" title="c3">q</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"> ( <font color="Maroon" title="c1">L</font> is  <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a>  &amp; <font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font> implies  <a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K9" title="FILTER_2:func.9">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K9" title="FILTER_2:func.9">#]</a></span>) is  <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:99"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
<font color="Maroon" title="c1">L</font> is  <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</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"> (  not <font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font> or  <a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K9" title="FILTER_2:func.9">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K9" title="FILTER_2:func.9">#]</a></span>) is  <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">P</font> = <span class="p1"><a href="filter_2.html#K9" title="FILTER_2:func.9">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K9" title="FILTER_2:func.9">#]</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">K</font> =  <a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K9" title="FILTER_2:func.9">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K9" title="FILTER_2:func.9">#]</a></span>);<br/>
<span class="kw">assume </span><a NAME="E2:99"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</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">  <a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K9" title="FILTER_2:func.9">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K9" title="FILTER_2:func.9">#]</a></span>) 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="c6">a9</font>, <font color="Maroon" title="c7">b9</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_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K9" title="FILTER_2:func.9">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K9" title="FILTER_2:func.9">#]</a></span>)</span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="filter_0.html#D6" 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">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K9" title="FILTER_2:func.9">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K9" title="FILTER_2:func.9">#]</a></span>)</span>)</span> st <br/>( <font color="Maroon" title="c6">a9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c7">b9</font> &amp; (  for <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K9" title="FILTER_2:func.9">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K9" title="FILTER_2:func.9">#]</a></span>)</span>)</span>  holds <br/>(  not <font color="Maroon" title="c6">a9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c7">b9</font> or <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive">b<sub>1</sub></font> ) ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c6">a9</font>, <font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c7">b9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T68" target="_self" onmouseover="rs('filter_2/T68')" onmouseout="rh()">Th68</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">c</font> = <font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font>;<br/>
<a NAME="E4:99"/><span class="lab"><font color="Green" title="E44">A3</font></span>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K9" title="FILTER_2:func.9">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K9" title="FILTER_2:func.9">#]</a></span>)) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K9" title="FILTER_2:func.9">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K9" title="FILTER_2:func.9">#]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T72" target="_self" onmouseover="rs('filter_2/T72')" onmouseout="rh()">Th72</a></span>;<br/>
<a NAME="E5:99"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="ref" href="filter_2.html#T62" target="_self" onmouseover="rs('filter_2/T62')" onmouseout="rh()">Th62</a></span>;<br/>
<a NAME="E6:99"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A4</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="lattices.html#D12" onmouseover="rs('lattices/D12')" onmouseout="rh()">LATTICES:def 12</a></span>;<br/>
<a NAME="E7:99"/><span class="lab"><font color="Green" title="E46">A5</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="filter_0.html#D7" onmouseover="rs('filter_0/D7')" onmouseout="rh()">FILTER_0:def 7</a></span>;<br/>
<a NAME="E8:99"/>
<font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E4:99"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="filter_2.html#T62" target="_self" onmouseover="rs('filter_2/T62')" onmouseout="rh()">Th62</a></span>;<br/>
<a NAME="E9:99"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="ref" href="filter_0.html#T6" onmouseover="rs('filter_0/T6')" onmouseout="rh()">FILTER_0:6</a></span>;<br/>
<a NAME="E10:99"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span></span>)</span></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T2" onmouseover="rs('filter_0/T2')" onmouseout="rh()">FILTER_0:2</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">d</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font>;<br/>
<a NAME="E11:99"/>
<font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T5" onmouseover="rs('lattices/T5')" onmouseout="rh()">LATTICES:5</a></span>;<br/>
<a NAME="E12:99"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="ref" href="filter_0.html#T7" onmouseover="rs('filter_0/T7')" onmouseout="rh()">FILTER_0:7</a>, <a class="ref" href="lattices.html#T6" onmouseover="rs('lattices/T6')" onmouseout="rh()">LATTICES:6</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">d9</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</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_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K9" title="FILTER_2:func.9">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K9" title="FILTER_2:func.9">#]</a></span>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E4:99"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="filter_2.html#T62" target="_self" onmouseover="rs('filter_2/T62')" onmouseout="rh()">Th62</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c12">d9</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="c6">a9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c12">d9</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c7">b9</font> &amp; (  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K9" title="FILTER_2:func.9">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K9" title="FILTER_2:func.9">#]</a></span>)</span>)</span>  holds <br/>(  not <font color="Maroon" title="c6">a9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c7">b9</font> or <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c12">d9</font> ) ) )</span><br/>

<a NAME="E14:99"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font></span>)</span> &amp; <font color="Maroon" title="c8">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c12">d9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T73" target="_self" onmouseover="rs('filter_2/T73')" onmouseout="rh()">Th73</a>, <a class="ref" href="lattices.html#D7" onmouseover="rs('lattices/D7')" onmouseout="rh()">LATTICES:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:99"/>
<font color="Maroon" title="c6">a9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c12">d9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:99"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="#E10:99"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="ref" href="filter_2.html#T74" target="_self" onmouseover="rs('filter_2/T74')" onmouseout="rh()">Th74</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">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K9" title="FILTER_2:func.9">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K9" title="FILTER_2:func.9">#]</a></span>)</span>)</span>  holds <br/>(  not <font color="Maroon" title="c6">a9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c7">b9</font> or <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c12">d9</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c13">e9</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_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K9" title="FILTER_2:func.9">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K9" title="FILTER_2:func.9">#]</a></span>)</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"> (  not <font color="Maroon" title="c6">a9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c13">e9</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c7">b9</font> or <font color="Maroon" title="c13">e9</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c12">d9</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">e</font> = <font color="Maroon" title="c13">e9</font>, <font color="Maroon" title="c15">ae</font> = <font color="Maroon" title="c6">a9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c13">e9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T68" target="_self" onmouseover="rs('filter_2/T68')" onmouseout="rh()">Th68</a></span>;<br/>
<a NAME="E17:99"/>
<font color="Maroon" title="c14">e</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E4:99"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="filter_2.html#T62" target="_self" onmouseover="rs('filter_2/T62')" onmouseout="rh()">Th62</a></span>;<br/>
<a NAME="E18:99"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A7</font></span>: 
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">e</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T4" onmouseover="rs('lattices/T4')" onmouseout="rh()">LATTICES:4</a></span>;<br/>
<span class="kw">assume </span><a NAME="E19:99"/>
<font color="Maroon" title="c6">a9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c13">e9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">b9</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">e9</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c12">d9</font></span><br/>

<a NAME="E20:99"/><span class="kw">then </span>
<font color="Maroon" title="c15">ae</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T74" target="_self" onmouseover="rs('filter_2/T74')" onmouseout="rh()">Th74</a></span>;<br/>
<a NAME="E21:99"/><span class="kw">then </span>
<font color="Maroon" title="c8">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c14">e</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T73" target="_self" onmouseover="rs('filter_2/T73')" onmouseout="rh()">Th73</a></span>;<br/>
<a NAME="E22:99"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A8</font></span>: 
<font color="Maroon" title="c14">e</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="filter_0.html#D7" onmouseover="rs('filter_0/D7')" onmouseout="rh()">FILTER_0:def 7</a></span>;<br/>
<a NAME="E23:99"/>
<font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c14">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E4:99"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="filter_2.html#T62" target="_self" onmouseover="rs('filter_2/T62')" onmouseout="rh()">Th62</a></span>;<br/>
<a NAME="E24:99"/><span class="kw">then </span>
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">e</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">p</font>
 
;<br/>
<a NAME="E25:99"/><span class="kw">then </span>
<font color="Maroon" title="c14">e</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:99"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="ref" href="filter_0.html#T1" onmouseover="rs('filter_0/T1')" onmouseout="rh()">FILTER_0:1</a></span>;<br/>
<a NAME="E26:99"/><span class="kw">then </span>
<font color="Maroon" title="c14">e</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:99"><span class="lab"><font color="Green" title="E48">A7</font></span></a>, <a class="ref" href="lattices.html#T9" onmouseover="rs('lattices/T9')" onmouseout="rh()">LATTICES:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:99"/>
<font color="Maroon" title="c13">e9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">d9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T74" target="_self" onmouseover="rs('filter_2/T74')" onmouseout="rh()">Th74</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/>


</div>
