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<b>let </b><font color="Maroon" title="c1">T</font> be  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel19.html#V2" title="WAYBEL19:attr.2">Lawson</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>;<br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a> <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a> <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c3">X</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">X</font>
;<br/>

<b>thus </b><a NAME="E2:61"/>
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">S</font> = <font color="Maroon" title="c2">S</font> as  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_5.html#NM3" title="WAYBEL_5:NM.3">CLSubFrame</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="waybel21.html#T18" target="_self" title="WAYBEL21:th.18">Th18</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">SL</font> being  <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a> <a href="waybel19.html#V2" title="WAYBEL19:attr.2">Lawson</a>  <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c4">S</font>;<br/>
<a NAME="E5:61"/><i><font color="Green" title="E39">A1</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">SL</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c5">SL</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">S</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">S</font> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E6:61"/><b>then </b><i><font color="Green" title="E40">A2</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c5">SL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12" title="PRE_TOPC:th.12">PRE_TOPC:12</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">f</font> =  <a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> <font color="Maroon" title="c5">SL</font>,<font color="Maroon" title="c1">T</font>;<br/>
<b>set </b><font color="Maroon" title="c7">f'</font> =  <a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> <font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">T</font>;<br/>
<a NAME="E7:61"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E8:61"/><b>then </b><i><font color="Green" title="E41">A3</font></i>: 
(  <a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> <font color="Maroon" title="c5">SL</font>,<font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">SL</font> &amp;  <a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> <font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">SL</font> )
 
<b>by </b><i><a class="txt" href="waybel21.html#E5:61"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="yellow_9.html#D1" title="YELLOW_9:def.1">YELLOW_9:def 1</a></i>;<br/>
<a NAME="E9:61"/><i><font color="Green" title="E42">A4</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c5">SL</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="ref" href="compts_1.html#T10" title="COMPTS_1:th.10">COMPTS_1:10</a></i>;<br/>
<a NAME="E10:61"/>
(  <a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> <font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">T</font> is <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a> &amp;  <a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> <font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">T</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> &amp;  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">T</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">T</font> #) )
 
<b>by </b><i><a class="ref" href="waybel21.html#T8" target="_self" title="WAYBEL21:th.8">Th8</a>, <a class="ref" href="waybel21.html#T10" target="_self" title="WAYBEL21:th.10">Th10</a></i>;<br/>
<a NAME="E11:61"/><b>then </b><i><font color="Green" title="E43">A5</font></i>: 
(  <a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> <font color="Maroon" title="c5">SL</font>,<font color="Maroon" title="c1">T</font> is <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a> &amp;  <a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> <font color="Maroon" title="c5">SL</font>,<font color="Maroon" title="c1">T</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> )
 
<b>by </b><i><a class="txt" href="waybel21.html#E5:61"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="waybel21.html#E8:61"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="waybel21.html#T6" target="_self" title="WAYBEL21:th.6">Th6</a></i>;<br/>
<a NAME="E12:61"/><b>then </b>
 <a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> <font color="Maroon" title="c5">SL</font>,<font color="Maroon" title="c1">T</font> is    <a href="waybel21.html#M1" title="WAYBEL21:mode.1">SemilatticeHomomorphism</a> of <font color="Maroon" title="c5">SL</font>,<font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="waybel21.html#T5" target="_self" title="WAYBEL21:th.5">Th5</a></i>;<br/>
<a NAME="E13:61"/><b>then </b>
 <a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> <font color="Maroon" title="c5">SL</font>,<font color="Maroon" title="c1">T</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel21.html#E11:61"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="waybel21.html#T46" target="_self" title="WAYBEL21:th.46">Th46</a></i>;<br/>
<a NAME="E14:61"/><b>then </b>
<span class="p1">(<span class="default"><a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> <font color="Maroon" title="c5">SL</font>,<font color="Maroon" title="c1">T</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c5">SL</font></span>)</span> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="txt" href="waybel21.html#E9:61"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="weierstr.html#T14" title="WEIERSTR:th.14">WEIERSTR:14</a></i>;<br/>
<a NAME="E15:61"/><b>then </b>
<font color="Maroon" title="c3">X</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="txt" href="waybel21.html#E6:61"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="waybel21.html#E8:61"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="funct_1.html#T162" title="FUNCT_1:th.162">FUNCT_1:162</a></i>;<br/>
<b>hence </b><a NAME="E16:61"/>
<font color="Maroon" title="c3">X</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="compts_1.html#T16" title="COMPTS_1:th.16">COMPTS_1:16</a></i>;<br/>


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