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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM6" title="WAYBEL_0:NM.6">Semilattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a> iff for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">J</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">J</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Olive" title="b1">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_2.html#K2" title="WAYBEL_2:func.2">FinSups</a> <font color="Olive" title="b3">f</font></span>)</span></span>)</span> )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">J</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">J</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Olive" title="b1">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_2.html#K2" title="WAYBEL_2:func.2">FinSups</a> <font color="Olive" title="b3">f</font></span>)</span></span>)</span> ) implies <font color="Maroon" title="c1">L</font> is <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:80_1"/>
<font color="Maroon" title="c1">L</font> is <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">J</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">J</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Olive" title="b1">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_2.html#K2" title="WAYBEL_2:func.2">FinSups</a> <font color="Olive" title="b3">f</font></span>)</span></span>)</span></span><br/><a NAME="E2:80_1"/><b>then </b>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">N</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="waybel_0.html#NM1" title="WAYBEL_0:NM.1">prenet</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">N</font> is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a> holds <br/><font color="Olive" title="b1">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Olive" title="b2">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <font color="Olive" title="b2">N</font>,<font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_2.html#T54" target="_self" title="WAYBEL_2:th.54">Th54</a></i>;<br/><b>hence </b><a NAME="E3:80_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">J</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">J</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Olive" title="b1">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_2.html#K2" title="WAYBEL_2:func.2">FinSups</a> <font color="Olive" title="b3">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_2.html#T47" target="_self" title="WAYBEL_2:th.47">Th47</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:80"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">J</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">J</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Olive" title="b1">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_2.html#K2" title="WAYBEL_2:func.2">FinSups</a> <font color="Olive" title="b3">f</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">L</font> is <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a></span><br/>

<a NAME="E3:80"/><b>then </b>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">N</font> being  <a href="waybel_0.html#NM1" title="WAYBEL_0:NM.1">prenet</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">N</font> is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a> holds <br/><font color="Olive" title="b1">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Olive" title="b2">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <font color="Olive" title="b2">N</font>,<font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_2.html#T48" target="_self" title="WAYBEL_2:th.48">Th48</a></i>;<br/>
<b>hence </b><a NAME="E4:80"/>
( <font color="Maroon" title="c1">L</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="waybel_2.html#V1" title="WAYBEL_2:attr.1">satisfying_MC</a> )
 <b>by </b><i><a class="ref" href="waybel_2.html#T42" target="_self" title="WAYBEL_2:th.42">Th42</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_2.html#D7" target="_self" title="WAYBEL_2:def.7">WAYBEL_2:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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