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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel_0.html#NM7" title="WAYBEL_0:NM.7">sup-Semilattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> is    <a href="waybel23.html#M1" title="WAYBEL23:mode.1">CLbasis</a> of <font color="Maroon" title="c1">L</font> implies <font color="Maroon" title="c1">L</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> )</span><br/>

<a NAME="E1:74"/><i><font color="Green" title="E37">A1</font></i>: 
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> holds <br/> ( not  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Olive" title="b1">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Olive" title="b1">x</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 
;<br/>
<b>assume </b><a NAME="E2:74"/><i><font color="Green" title="E38">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> is    <a href="waybel23.html#M1" title="WAYBEL23:mode.1">CLbasis</a> of <font color="Maroon" title="c1">L</font>
 ; <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_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a></span><br/>

<b>reconsider </b><font color="Maroon" title="c2">C</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="waybel23.html#T43" target="_self" title="WAYBEL23:th.43">Th43</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <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"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">x</font></span>)</span></span><br/><a NAME="E1:74_1"/>
<font color="Maroon" title="c3">x</font> <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">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:74"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="waybel23.html#D7" target="_self" title="WAYBEL23:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E2:74_1"/>
<font color="Maroon" title="c3">x</font> <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"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel23.html#T1" target="_self" title="WAYBEL23:th.1">Th1</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>
<a NAME="E5:74"/><b>then </b>
<font color="Maroon" title="c1">L</font> is <a href="waybel_8.html#V1" title="WAYBEL_8:attr.1">satisfying_axiom_K</a>
 
<b>by </b><i><a class="ref" href="waybel_8.html#D3" title="WAYBEL_8:def.3">WAYBEL_8:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:74"/>
<font color="Maroon" title="c1">L</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>
 <b>by </b><i><a class="txt" href="#E1:74"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="waybel_8.html#D4" title="WAYBEL_8:def.4">WAYBEL_8:def 4</a></i>; <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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