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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a> <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font> is <a href="yellow_0.html#V6" title="YELLOW_0:attr.6">join-inheriting</a> &amp;  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</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="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">L</font> implies  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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> )</span><br/><b>assume </b><b>that </b><br/><a NAME="E1:5_1"/><i><font color="Green" title="E2">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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>
 <b>and </b><br/><a NAME="E2:5_1"/><i><font color="Green" title="E3">A2</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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>
 <b>and </b><br/><a NAME="E3:5_1"/><i><font color="Green" title="E4">A3</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<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">  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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></span><br/><a NAME="E4:5_1"/><i><font color="Green" title="E5">A4</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> )
 <b>by </b><i><a class="txt" href="#E3:5_1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="yellow_0.html#T18" title="YELLOW_0:th.18">YELLOW_0:18</a></i>;<br/><a NAME="E5:5_1"/>
<font color="Maroon" title="c2">x</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 <b>by </b><i><a class="txt" href="#E1:5_1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="waybel_8.html#D1" target="_self" title="WAYBEL_8:def.1">Def1</a></i>;<br/><a NAME="E6:5_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#D2" title="WAYBEL_3:def.2">WAYBEL_3:def 2</a></i>;<br/><a NAME="E7:5_1"/><b>then </b><i><font color="Green" title="E6">A5</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="#E4:5_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="waybel_3.html#T2" title="WAYBEL_3:th.2">WAYBEL_3:2</a></i>;<br/><a NAME="E8:5_1"/>
<font color="Maroon" title="c3">y</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 <b>by </b><i><a class="txt" href="#E2:5_1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="waybel_8.html#D1" target="_self" title="WAYBEL_8:def.1">Def1</a></i>;<br/><a NAME="E9:5_1"/><b>then </b>
<font color="Maroon" title="c3">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#D2" title="WAYBEL_3:def.2">WAYBEL_3:def 2</a></i>;<br/><a NAME="E10:5_1"/><b>then </b>
<font color="Maroon" title="c3">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="#E4:5_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="waybel_3.html#T2" title="WAYBEL_3:th.2">WAYBEL_3:2</a></i>;<br/><a NAME="E11:5_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="#E7:5_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="waybel_3.html#T3" title="WAYBEL_3:th.3">WAYBEL_3:3</a></i>;<br/><a NAME="E12:5_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 <b>by </b><i><a class="ref" href="waybel_3.html#D2" title="WAYBEL_3:def.2">WAYBEL_3:def 2</a></i>;<br/><a NAME="E13:5_1"/><b>then </b>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 <b>by </b><i><a class="ref" href="yellow_0.html#T41" title="YELLOW_0:th.41">YELLOW_0:41</a></i>;<br/><b>hence </b><a NAME="E14:5_1"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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>
 <b>by </b><i><a class="ref" href="waybel_8.html#D1" target="_self" title="WAYBEL_8:def.1">Def1</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>hence </b><a NAME="E2:5"/>
 <a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font> is <a href="yellow_0.html#V6" title="YELLOW_0:attr.6">join-inheriting</a>
 <b>by </b><i><a class="ref" href="yellow_0.html#D17" title="YELLOW_0:def.17">YELLOW_0:def 17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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></span><br/>

<a NAME="E3:5"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T15" title="WAYBEL_3:th.15">WAYBEL_3:15</a></i>;<br/>
<b>hence </b><a NAME="E4:5"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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>
 <b>by </b><i><a class="ref" href="waybel_8.html#D1" target="_self" title="WAYBEL_8:def.1">Def1</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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