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<div class="add">

<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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">I</font>, <font color="Olive" title="b2">J</font> being   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">L</font> holds  <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">J</font></span>)</span></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b3">r</font> where <font color="Olive" title="b3">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">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 <br/>( <font color="Olive" title="b3">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b4">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b5">q</font> &amp; <font color="Olive" title="b4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">I</font> &amp; <font color="Olive" title="b5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">J</font> ) </span> } </span> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">I</font>, <font color="Maroon" title="c3">J</font> be   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font></span>)</span></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">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 <br/>( <font color="Olive" title="b1">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> ) </span> } </span> </span><br/>





<a NAME="E1:69"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c3">J</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font>
 
;<br/>
<a NAME="E2:69"/>
( <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font></span>)</span></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> &amp; <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T36" target="_self" title="FILTER_2:th.36">Th37</a></span>;<br/>
<a NAME="E3:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font></span>)</span></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r9</font> where <font color="Olive" title="b1">r9</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a></span>)</span> :  ex <font color="Olive" title="b2">p9</font>, <font color="Olive" title="b3">q9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a></span>)</span> st <br/>( <font color="Olive" title="b2">p9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b3">q9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">r9</font> &amp; <font color="Olive" title="b2">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b3">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="filter_0.html#T35" title="FILTER_0:th.35">FILTER_0:35</a></span>;<br/>
<span class="kw">thus </span><a NAME="E4:69"/>
<span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font></span>)</span></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">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 <br/>( <font color="Olive" title="b1">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> ) </span> } </span> 
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">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 <br/>( <font color="Olive" title="b1">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font></span>)</span></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font></span>)</span></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">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 <br/>( <font color="Olive" title="b1">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> ) </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:69_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font></span>)</span></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">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 <br/>( <font color="Olive" title="b1">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> ) </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">r9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:69_1"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r9</font>
 <span class="kw">and </span><br/><a NAME="E4:69_1"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
 ex <font color="Olive" title="b1">p9</font>, <font color="Olive" title="b2">q9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a></span>)</span> st <br/>( <font color="Olive" title="b1">p9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">q9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">r9</font> &amp; <font color="Olive" title="b1">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b2">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69"><span class="lab"><font color="Green" title="E26">A2</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">p9</font>, <font color="Maroon" title="c7">q9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E6:69_1"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
<font color="Maroon" title="c6">p9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">q9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">r9</font>
 <span class="kw">and </span><br/><a NAME="E7:69_1"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
( <font color="Maroon" title="c6">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c7">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a></span>;<br/>
<a NAME="E8:69_1"/><span class="lab"><font color="Green" title="E31">A7</font></span>: 
<font color="Maroon" title="c6">p9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="filter_2.html#K2" title="FILTER_2:func.2">.:</a> <font color="Maroon" title="c6">p9</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="filter_2.html#K2" title="FILTER_2:func.2">.:</a> <font color="Maroon" title="c7">q9</font></span>)</span>
 
;<br/>
<a NAME="E9:69_1"/>
 <a href="filter_2.html#K2" title="FILTER_2:func.2">.:</a> <font color="Maroon" title="c5">r9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="filter_2.html#K2" title="FILTER_2:func.2">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">q9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="lattice2.html#T39" title="LATTICE2:th.39">LATTICE2:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:69_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">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 <br/>( <font color="Olive" title="b1">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E7:69_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E8:69_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">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 <br/>( <font color="Olive" title="b1">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> ) </span> } </span>  or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font></span>)</span></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E5:69"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">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 <br/>( <font color="Olive" title="b1">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> ) </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font></span>)</span></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E7:69"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font>
 <span class="kw">and </span><br/><a NAME="E8:69"/><span class="lab"><font color="Green" title="E28">A9</font></span>: 
 ex <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 <br/>( <font color="Maroon" title="c5">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E10:69"/><span class="lab"><font color="Green" title="E29">A10</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">q</font>
 <span class="kw">and </span><br/><a NAME="E11:69"/><span class="lab"><font color="Green" title="E30">A11</font></span>: 
( <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:69"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>;<br/>
<a NAME="E12:69"/><span class="lab"><font color="Green" title="E31">A12</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a></span>)</span>
 
;<br/>
<a NAME="E13:69"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">q</font></span>)</span> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a>  <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">r</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:69"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="lattice2.html#T38" title="LATTICE2:th.38">LATTICE2:38</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:69"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font></span>)</span></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E7:69"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E11:69"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="txt" href="#E12:69"><span class="lab"><font color="Green" title="E31">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
