<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">N</font> be   <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>   <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">N</font></span>)</span> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  implies <font color="Maroon" title="c1">N</font> is  <a href="yellow13.html#V2" title="YELLOW13:attr.2">topological_semilattice</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:37"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">N</font></span>)</span> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">N</font> is  <a href="yellow13.html#V2" title="YELLOW13:attr.2">topological_semilattice</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">NN</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 <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c1">N</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>;<br/>
<a NAME="E3:37"/>
<font color="Maroon" title="c1">N</font> is    <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#T44" title="YELLOW_9:th.44">YELLOW_9:44</a></span>;<br/>
<a NAME="E4:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A2</font></span>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">NN</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">NN</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c1">N</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="waybel30.html#T20" target="_self" title="WAYBEL30:th.20">Th20</a></span>;<br/>
<a NAME="E5:37"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
 <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="c2">NN</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">NN</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 <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c1">N</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c1">N</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">h</font> =  <a href="waybel_2.html#K4" title="WAYBEL_2:func.4">inf_op</a> <font color="Maroon" title="c1">N</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">NN</font>,<font color="Maroon" title="c1">N</font> ;<br/>
<a NAME="E7:37"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
 <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">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">N</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">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">N</font> #)
 
;<br/>
<a NAME="E8:37"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<font color="Maroon" title="c3">h</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">NN</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c4">X</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  or <font color="Maroon" title="c3">h</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c4">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_1"/>
( not <font color="Maroon" title="c4">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c4">X</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">h</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c4">X</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">X1</font> = <font color="Maroon" title="c4">X</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c1">N</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="waybel_0.html#T3" title="WAYBEL_0:th.3">WAYBEL_0:3</a></span>;<br/>
<a NAME="E3:37_1"/>
 <a href="waybel_2.html#K4" title="WAYBEL_2:func.4">inf_op</a> <font color="Maroon" title="c1">N</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c5">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:37_1"/>
<font color="Maroon" title="c3">h</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:37"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="waybel_0.html#T65" title="WAYBEL_0:th.65">WAYBEL_0:65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:37"/><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<font color="Maroon" title="c3">h</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">NN</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_0.html#D32" title="WAYBEL_0:def.32">WAYBEL_0:def 32</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">h</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <font color="Maroon" title="c4">X</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">X1</font> = <font color="Maroon" title="c4">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c1">N</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37"><span class="lab"><font color="Green" title="E22">A3</font></span></a></span>;<br/>
<a NAME="E2:37_2"/>
 <a href="waybel_2.html#K4" title="WAYBEL_2:func.4">inf_op</a> <font color="Maroon" title="c1">N</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <font color="Maroon" title="c5">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#D32" title="WAYBEL_0:def.32">WAYBEL_0:def 32</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:37_2"/>
<font color="Maroon" title="c3">h</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:37"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="waybel_0.html#T65" title="WAYBEL_0:th.65">WAYBEL_0:65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:37"/><span class="kw">then </span>
<font color="Maroon" title="c3">h</font> is    <a href="waybel21.html#M1" title="WAYBEL21:mode.1">SemilatticeHomomorphism</a> of <font color="Maroon" title="c2">NN</font>,<font color="Maroon" title="c1">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel21.html#T5" title="WAYBEL21:th.5">WAYBEL21:5</a></span>;<br/>
<a NAME="E11:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A7</font></span>: 
<font color="Maroon" title="c3">h</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:37"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E9:37"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="waybel21.html#T46" title="WAYBEL21:th.46">WAYBEL21:46</a></span>;<br/>
<a NAME="E12:37"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">N</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">N</font> #)
 
;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c1">N</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>,<font color="Maroon" title="c1">N</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow13.html#D5" title="YELLOW13:def.5">YELLOW13:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <a href="funct_2.html#R1" title="FUNCT_2:pred.1">R18</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c1">N</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c1">N</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c4">g</font>, <a href="waybel_2.html#K4" title="WAYBEL_2:func.4">inf_op</a> <font color="Maroon" title="c1">N</font>) or <font color="Maroon" title="c4">g</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E13:37"/>
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_2.html#K4" title="WAYBEL_2:func.4">inf_op</a> <font color="Maroon" title="c1">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">g</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> </span><br/>

<span class="kw">hence </span><a NAME="E14:37"/>
<font color="Maroon" title="c4">g</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E11:37"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E12:37"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="ref" href="yellow12.html#T36" title="YELLOW12:th.36">YELLOW12:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
