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<span class="kw">set </span><font color="Maroon" title="c3">I0</font> =  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</font>;<br/>
<a NAME="E1:50_2_1"/><span class="lab"><font color="Green" title="E22">CC</font></span>: 
 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="Olive" title="b1">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</font> holds <br/><font color="Olive" title="b2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">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="c4">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c4">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</font> implies <font color="Maroon" title="c5">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_2_1_1"/><span class="lab"><font color="Green" title="E23">C0</font></span>: 
( <font color="Maroon" title="c4">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c4">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x1</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="E3:50_2_1_1"/><span class="lab"><font color="Green" title="E24">C1</font></span>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> )
 ;<br/>
<a NAME="E4:50_2_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_2_1_1"><span class="lab"><font color="Green" title="E23">C0</font></span></a>, <a class="ref" href="filter_0.html#T1" onmouseover="rs('filter_0/T1')" onmouseout="rh()">FILTER_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:50_2_1_1"/>
<font color="Maroon" title="c5">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_2_1_1"><span class="lab"><font color="Green" title="E24">C1</font></span></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><span class="kw">end;</span></div>
<a NAME="E2:50_2_1"/><span class="lab"><font color="Green" title="E23">C2</font></span>: 
 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="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</font> holds <br/><font color="Olive" title="b1">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">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="c4">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c5">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</font> implies <font color="Maroon" title="c4">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_2_1_2"/><span class="lab"><font color="Green" title="E24">d0</font></span>: 
( <font color="Maroon" title="c4">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c5">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</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="c4">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x1</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="E3:50_2_1_2"/><span class="lab"><font color="Green" title="E25">d1</font></span>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">x2</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="E5:50_2_1_2"/><span class="lab"><font color="Green" title="E26">d2</font></span>: 
( <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_2_1_2"><span class="lab"><font color="Green" title="E24">d0</font></span></a></span>;<br/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">x2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">x2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="latticea.html#E39"><span class="lab"><font color="Green" title="E21">LemDistr</font></span></a></span>
<br/>.= 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_2_1_2"><span class="lab"><font color="Green" title="E25">d1</font></span></a>, <a class="txt" href="#E5:50_2_1_2"><span class="lab"><font color="Green" title="E26">d2</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:50_2_1_2"/>
<font color="Maroon" title="c4">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_2_1_2"><span class="lab"><font color="Green" title="E25">d1</font></span></a>, <a class="txt" href="#E5:50_2_1_2"><span class="lab"><font color="Green" title="E26">d2</font></span></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><span class="kw">end;</span></div>
<a NAME="E3:50_2_1"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font>
 
;<br/>
<a NAME="E4:50_2_1"/><span class="kw">then </span>
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E5:50_2_1"/>
(  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</font> is  <a href="lattices.html#V19" title="LATTICES:attr.19">final</a>  &amp;  not  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</font> is  <a href="lattices.html#V20" title="LATTICES:attr.20">meet-closed</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_2_1"><span class="lab"><font color="Green" title="E22">CC</font></span></a>, <a class="txt" href="#E2:50_2_1"><span class="lab"><font color="Green" title="E23">C2</font></span></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>
