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<b>let </b><font color="Maroon">T</font> be  <a href="waybel_3.html#V3">continuous</a> <a href="lattice3.html#V3">complete</a> <a href="waybel19.html#V2">Lawson</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a> <a href="yellow_0.html#V7">infs-inheriting</a> <a href="waybel_0.html#V4">directed-sups-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">T</font>;<br/>



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

<b>thus </b><a NAME="E2:61"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">S</font> = <font color="Maroon">S</font> as  <a href="lattice3.html#V3">complete</a> <a href="waybel_5.html#NM3">CLSubFrame</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">SL</font> being  <a href="pre_topc.html#V2">correct</a> <a href="waybel19.html#V2">Lawson</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">S</font>;<br/>
<a NAME="E5:61"/><i><font color="Green">E26</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SL</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">SL</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E6:61"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">SL</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<b>set </b><font color="Maroon">f</font> =  <a href="yellow_9.html#K1">incl</a> <font color="Maroon">SL</font>,<font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">f'</font> =  <a href="yellow_9.html#K1">incl</a> <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>
<a NAME="E7:61"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E8:61"/><b>then </b><i><font color="Green">E31</font></i>: 
(  <a href="yellow_9.html#K1">incl</a> <font color="Maroon">SL</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SL</font> &amp;  <a href="yellow_9.html#K1">incl</a> <font color="Maroon">S</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SL</font> )
 
<b>by </b><i><a class="txt" href="waybel21.html#E11:61"><i><font color="Green">E39</font></i></a>, <a class="ref" href="yellow_9.html#D1">YELLOW_9:def 1</a></i>;<br/>
<a NAME="E9:61"/><i><font color="Green">E38</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">SL</font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="ref" href="compts_1.html#T10">COMPTS_1:10</a></i>;<br/>
<a NAME="E10:61"/>
(  <a href="yellow_9.html#K1">incl</a> <font color="Maroon">S</font>,<font color="Maroon">T</font> is <a href="waybel_0.html#V17">infs-preserving</a> &amp;  <a href="yellow_9.html#K1">incl</a> <font color="Maroon">S</font>,<font color="Maroon">T</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font> #) )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:61"/><b>then </b><i><font color="Green">E39</font></i>: 
(  <a href="yellow_9.html#K1">incl</a> <font color="Maroon">SL</font>,<font color="Maroon">T</font> is <a href="waybel_0.html#V17">infs-preserving</a> &amp;  <a href="yellow_9.html#K1">incl</a> <font color="Maroon">SL</font>,<font color="Maroon">T</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> )
 
<b>by </b><i><a class="txt" href="waybel21.html#E11:61"><i><font color="Green">E39</font></i></a>, , </i>;<br/>
<a NAME="E12:61"/><b>then </b>
 <a href="yellow_9.html#K1">incl</a> <font color="Maroon">SL</font>,<font color="Maroon">T</font> is    <a href="waybel21.html#M1" target="_self">SemilatticeHomomorphism</a> of <font color="Maroon">SL</font>,<font color="Maroon">T</font>
 
<b>by </b><i/>;<br/>
<a NAME="E13:61"/><b>then </b>
 <a href="yellow_9.html#K1">incl</a> <font color="Maroon">SL</font>,<font color="Maroon">T</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:61"/><b>then </b>
<span class="p1">(<span class="default"><a href="yellow_9.html#K1">incl</a> <font color="Maroon">SL</font>,<font color="Maroon">T</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">SL</font></span>)</span> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i>, <a class="ref" href="weierstr.html#T14">WEIERSTR:14</a></i>;<br/>
<a NAME="E15:61"/><b>then </b>
<font color="Maroon">X</font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T162">FUNCT_1:162</a></i>;<br/>
<b>hence </b><a NAME="E16:61"/>
<font color="Maroon">X</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="compts_1.html#T16">COMPTS_1:16</a></i>;<br/>


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