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<span class="kw">let </span><font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c2">L2</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">L1</font> is  <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a>  &amp; <font color="Maroon" title="c2">L2</font> is  <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a>  ) iff <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> is  <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a>  )</span><br/>



<span class="kw">thus </span><a NAME="E1:67"/>
( <font color="Maroon" title="c1">L1</font> is  <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a>  &amp; <font color="Maroon" title="c2">L2</font> is  <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a>  implies <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> is  <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> is  <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a>  implies ( <font color="Maroon" title="c1">L1</font> is  <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a>  &amp; <font color="Maroon" title="c2">L2</font> is  <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a>  ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:67_1"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">q1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>  ex <font color="Olive" title="b3">r1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> st <br/>( <font color="Olive" title="b1">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b3">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">q1</font> &amp; ( for <font color="Olive" title="b4">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>  st <font color="Olive" title="b1">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b4">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">q1</font> holds <br/><font color="Olive" title="b4">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b3">r1</font> ) )
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="filter_0.html#D7" title="FILTER_0:def.7">FILTER_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">L2</font> is  <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a>  or <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</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="E2:67_1"/><span class="lab"><font color="Green" title="E44">A2</font></span>: 
for <font color="Olive" title="b1">p2</font>, <font color="Olive" title="b2">q2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>  ex <font color="Olive" title="b3">r2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font> st <br/>( <font color="Olive" title="b1">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b3">r2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">q2</font> &amp; ( for <font color="Olive" title="b4">s2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>  st <font color="Olive" title="b1">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b4">s2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">q2</font> holds <br/><font color="Olive" title="b4">s2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b3">r2</font> ) )
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="filter_0.html#D7" title="FILTER_0:def.7">FILTER_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</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="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="filter_0.html#D7" title="FILTER_0:def.7">FILTER_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> st <br/>( <font color="Maroon" title="c3">a</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="c4">b</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"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>  holds <br/>(  not <font color="Maroon" title="c3">a</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="c4">b</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">consider </span><font color="Maroon" title="c5">p1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c6">p2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><span class="kw"> such that </span><br/><a NAME="E4:67_1"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">q1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c8">q2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><span class="kw"> such that </span><br/><a NAME="E6:67_1"/><span class="lab"><font color="Green" title="E46">A4</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c8">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">r2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><span class="kw"> such that </span><br/><a NAME="E8:67_1"/><span class="lab"><font color="Green" title="E47">A5</font></span>: 
<font color="Maroon" title="c6">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">r2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">q2</font>
 <span class="kw">and </span><br/><a NAME="E9:67_1"/><span class="lab"><font color="Green" title="E48">A6</font></span>: 
for <font color="Olive" title="b1">s2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>  st <font color="Maroon" title="c6">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">s2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">q2</font> holds <br/><font color="Olive" title="b1">s2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_1"><span class="lab"><font color="Green" title="E44">A2</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">r1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font><span class="kw"> such that </span><br/><a NAME="E11:67_1"/><span class="lab"><font color="Green" title="E49">A7</font></span>: 
<font color="Maroon" title="c5">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">q1</font>
 <span class="kw">and </span><br/><a NAME="E12:67_1"/><span class="lab"><font color="Green" title="E50">A8</font></span>: 
for <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>  st <font color="Maroon" title="c5">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">q1</font> holds <br/><font color="Olive" title="b1">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1"><span class="lab"><font color="Green" title="E43">A1</font></span></a></span>;<br/>
<span class="kw">take </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c4">b</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"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>  holds <br/>(  not <font color="Maroon" title="c3">a</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="c4">b</font> or <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> ) ) )</span><br/>

<a NAME="E13:67_1"/>
<font color="Maroon" title="c3">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">r1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">r2</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_1"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="filter_1.html#T22" target="_self" title="FILTER_1:th.22">Th22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:67_1"/>
<font color="Maroon" title="c3">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67_1"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E11:67_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E8:67_1"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="ref" href="filter_1.html#T37" target="_self" title="FILTER_1:th.37">Th37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>  holds <br/>(  not <font color="Maroon" title="c3">a</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="c4">b</font> or <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c11">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c11">d</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c4">b</font> or <font color="Maroon" title="c11">d</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c12">s1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c13">s2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><span class="kw"> such that </span><br/><a NAME="E16:67_1"/><span class="lab"><font color="Green" title="E51">A9</font></span>: 
<font color="Maroon" title="c11">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c12">s1</font>,<font color="Maroon" title="c13">s2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></span>;<br/>
<span class="kw">assume </span><a NAME="E17:67_1"/>
<font color="Maroon" title="c3">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c11">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">d</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span></span><br/>

<a NAME="E18:67_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A10</font></span>: 
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c12">s1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c13">s2</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_1"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E16:67_1"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="ref" href="filter_1.html#T22" target="_self" title="FILTER_1:th.22">Th22</a></span>;<br/>
<a NAME="E19:67_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c13">s2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67_1"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="filter_1.html#T37" target="_self" title="FILTER_1:th.37">Th37</a></span>;<br/>
<a NAME="E20:67_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A11</font></span>: 
<font color="Maroon" title="c13">s2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">r2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:67_1"><span class="lab"><font color="Green" title="E48">A6</font></span></a></span>;<br/>
<a NAME="E21:67_1"/>
<font color="Maroon" title="c5">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c12">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67_1"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E18:67_1"><span class="lab"><font color="Green" title="E52">A10</font></span></a>, <a class="ref" href="filter_1.html#T37" target="_self" title="FILTER_1:th.37">Th37</a></span>;<br/>
<a NAME="E22:67_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:67_1"><span class="lab"><font color="Green" title="E50">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:67_1"/>
<font color="Maroon" title="c11">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:67_1"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="txt" href="#E20:67_1"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="ref" href="filter_1.html#T37" target="_self" title="FILTER_1:th.37">Th37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:67"/><span class="lab"><font color="Green" title="E43">A12</font></span>: 
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 <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>  ex <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> st <br/>( <font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b3">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">b</font> &amp; ( for <font color="Olive" title="b4">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>  st <font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b4">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b4">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b3">c</font> ) )
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="filter_0.html#D7" title="FILTER_0:def.7">FILTER_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">L1</font> is  <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a>  &amp; <font color="Maroon" title="c2">L2</font> is  <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E3:67"/>
for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">q1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>  ex <font color="Olive" title="b3">r1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> st <br/>( <font color="Olive" title="b1">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b3">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">q1</font> &amp; ( for <font color="Olive" title="b4">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>  st <font color="Olive" title="b1">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b4">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">q1</font> holds <br/><font color="Olive" title="b4">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b3">r1</font> ) )
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="filter_0.html#D7" title="FILTER_0:def.7">FILTER_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">L2</font> is  <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c3">p2</font>, <font color="Maroon" title="c4">q2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">p1</font>, <font color="Maroon" title="c6">q1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <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">L1</font> st <br/>( <font color="Maroon" title="c5">p1</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="c6">q1</font> &amp; ( for <font color="Olive" title="b2">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>  st <font color="Maroon" title="c5">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">q1</font> holds <br/><font color="Olive" title="b2">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">r1</font> ) )</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c7">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span><span class="kw"> such that </span><br/><a NAME="E3:67_2"/><span class="lab"><font color="Green" title="E44">A13</font></span>: 
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c4">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:67_2"/><span class="lab"><font color="Green" title="E45">A14</font></span>: 
for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>  st <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c4">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> holds <br/><font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E43">A12</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">r1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c9">r2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><span class="kw"> such that </span><br/><a NAME="E6:67_2"/><span class="lab"><font color="Green" title="E46">A15</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c8">r1</font>,<font color="Maroon" title="c9">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">r1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">q1</font> &amp; ( for <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>  st <font color="Maroon" title="c5">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">q1</font> holds <br/><font color="Olive" title="b1">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">r1</font> ) )</span><br/>

<a NAME="E7:67_2"/><span class="lab"><font color="Green" title="E47">A16</font></span>: 
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">r1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">r2</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67_2"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="ref" href="filter_1.html#T22" target="_self" title="FILTER_1:th.22">Th22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:67_2"/>
<font color="Maroon" title="c5">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_2"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="ref" href="filter_1.html#T37" target="_self" title="FILTER_1:th.37">Th37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>  st <font color="Maroon" title="c5">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">q1</font> holds <br/><font color="Olive" title="b1">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">r1</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c10">s1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">q1</font> implies <font color="Maroon" title="c10">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">r1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E9:67_2"/><span class="lab"><font color="Green" title="E48">A17</font></span>: 
<font color="Maroon" title="c5">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">q1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">r1</font></span><br/>

<a NAME="E10:67_2"/>
<font color="Maroon" title="c3">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">r2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_2"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E7:67_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="ref" href="filter_1.html#T37" target="_self" title="FILTER_1:th.37">Th37</a></span>;<br/>
<a NAME="E11:67_2"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">s1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">r2</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c4">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:67_2"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="ref" href="filter_1.html#T37" target="_self" title="FILTER_1:th.37">Th37</a></span>;<br/>
<a NAME="E12:67_2"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c10">s1</font>,<font color="Maroon" title="c9">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c4">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T22" target="_self" title="FILTER_1:th.22">Th22</a></span>;<br/>
<a NAME="E13:67_2"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c10">s1</font>,<font color="Maroon" title="c9">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_2"><span class="lab"><font color="Green" title="E45">A14</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:67_2"/>
<font color="Maroon" title="c10">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67_2"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="ref" href="filter_1.html#T37" target="_self" title="FILTER_1:th.37">Th37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">consider </span><font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">q1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">p2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="filter_0.html#D7" title="FILTER_0:def.7">FILTER_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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 <font color="Maroon" title="c2">L2</font>  ex <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 <font color="Maroon" title="c2">L2</font> st <br/>( <font color="Maroon" title="c5">p2</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="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</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 <font color="Maroon" title="c2">L2</font>  holds <br/>(  not <font color="Maroon" title="c5">p2</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive">b<sub>3</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>3</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive">b<sub>2</sub></font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">q2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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 <font color="Maroon" title="c2">L2</font> st <br/>( <font color="Maroon" title="c5">p2</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="c6">q2</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 <font color="Maroon" title="c2">L2</font>  holds <br/>(  not <font color="Maroon" title="c5">p2</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="c6">q2</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">consider </span><font color="Maroon" title="c7">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span><span class="kw"> such that </span><br/><a NAME="E6:67"/><span class="lab"><font color="Green" title="E44">A18</font></span>: 
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c6">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:67"/><span class="lab"><font color="Green" title="E45">A19</font></span>: 
for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>  st <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c6">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> holds <br/><font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E43">A12</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">r1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c9">r2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><span class="kw"> such that </span><br/><a NAME="E9:67"/><span class="lab"><font color="Green" title="E46">A20</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c8">r1</font>,<font color="Maroon" title="c9">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">r2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">p2</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c9">r2</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">q2</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 <font color="Maroon" title="c2">L2</font>  holds <br/>(  not <font color="Maroon" title="c5">p2</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="c6">q2</font> or <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c9">r2</font> ) ) )</span><br/>

<a NAME="E10:67"/><span class="lab"><font color="Green" title="E47">A21</font></span>: 
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">r1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">r2</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:67"><span class="lab"><font color="Green" title="E46">A20</font></span></a>, <a class="ref" href="filter_1.html#T22" target="_self" title="FILTER_1:th.22">Th22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:67"/>
<font color="Maroon" title="c5">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">r2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="ref" href="filter_1.html#T37" target="_self" title="FILTER_1:th.37">Th37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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 <font color="Maroon" title="c2">L2</font>  holds <br/>(  not <font color="Maroon" title="c5">p2</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="c6">q2</font> or <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c9">r2</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c10">s2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">p2</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c10">s2</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">q2</font> or <font color="Maroon" title="c10">s2</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c9">r2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E12:67"/><span class="lab"><font color="Green" title="E48">A22</font></span>: 
<font color="Maroon" title="c5">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">s2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">q2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">s2</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c9">r2</font></span><br/>

<a NAME="E13:67"/>
<font color="Maroon" title="c3">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E47">A21</font></span></a>, <a class="ref" href="filter_1.html#T37" target="_self" title="FILTER_1:th.37">Th37</a></span>;<br/>
<a NAME="E14:67"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">r1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">s2</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c6">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:67"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="ref" href="filter_1.html#T37" target="_self" title="FILTER_1:th.37">Th37</a></span>;<br/>
<a NAME="E15:67"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c8">r1</font>,<font color="Maroon" title="c10">s2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c6">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T22" target="_self" title="FILTER_1:th.22">Th22</a></span>;<br/>
<a NAME="E16:67"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c8">r1</font>,<font color="Maroon" title="c10">s2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:67"><span class="lab"><font color="Green" title="E45">A19</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:67"/>
<font color="Maroon" title="c10">s2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:67"><span class="lab"><font color="Green" title="E46">A20</font></span></a>, <a class="ref" href="filter_1.html#T37" target="_self" title="FILTER_1:th.37">Th37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
