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

<b>let </b><font color="Maroon">N</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_2.html#V2">meet-continuous</a> <a href="waybel19.html#V2">Lawson</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/>

<b>assume </b><a NAME="E1:37"/><i><font color="Green">E32</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">N</font></span>)</span> is <a href="waybel_3.html#V3">continuous</a>
 ;<br/>

<b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">N</font></span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">N</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow13.html#D5">YELLOW13:def 5</a><br/>

<b>assume </b><a NAME="E2:37"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="waybel_2.html#K4">inf_op</a> <font color="Maroon">N</font>
 ;<br/>

<b>consider </b><font color="Maroon">NN</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 <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">N</font></span><a href="yellow_3.html#K3">:]</a></span>;<br/>
<a NAME="E4:37"/><i><font color="Green">E35</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">NN</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">NN</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 <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">N</font></span><a href="yellow_3.html#K3">:]</a></span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">N</font></span><a href="yellow_3.html#K3">:]</a></span> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">h</font> =  <a href="waybel_2.html#K4">inf_op</a> <font color="Maroon">N</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">NN</font>,<font color="Maroon">N</font> ;<br/>
<a NAME="E6:37"/><i><font color="Green">E36</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">N</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">N</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">N</font> #)
 
;<br/>
<a NAME="E7:37"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">h</font> is <a href="waybel_0.html#V17">infs-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">NN</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a><br/>

<b>reconsider </b><font color="Maroon">X1</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">N</font></span><a href="yellow_3.html#K3">:]</a></span> <b>by </b><i/>;<br/>
<a NAME="E2:37_1"/>
 <a href="waybel_2.html#K4">inf_op</a> <font color="Maroon">N</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">X1</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a></i>;<br/>
<b>hence </b><a NAME="E3:37_1"/>
<font color="Maroon">h</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#T65">WAYBEL_0:65</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:37"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">h</font> is    <a href="waybel21.html#M1">SemilatticeHomomorphism</a> of <font color="Maroon">NN</font>,<font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="waybel21.html#T5">WAYBEL21:5</a></i>;<br/>
<a NAME="E9:37"/>
<font color="Maroon">h</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">NN</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a><br/>

<b>assume </b><a NAME="E1:37_2"/>
( not <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">X</font> is <a href="waybel_0.html#V1">directed</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">X1</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">N</font></span><a href="yellow_3.html#K3">:]</a></span> <b>by </b><i>, <a class="ref" href="waybel_0.html#T3">WAYBEL_0:3</a></i>;<br/>
<a NAME="E3:37_2"/>
 <a href="waybel_2.html#K4">inf_op</a> <font color="Maroon">N</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">X1</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/>
<b>hence </b><a NAME="E4:37_2"/>
<font color="Maroon">h</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#T65">WAYBEL_0:65</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:37"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, , <a class="ref" href="waybel21.html#T46">WAYBEL21:46</a></i>;<br/>
<a NAME="E11:37"/><i><font color="Green">E49</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">N</font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">N</font> #)
 
;<br/>
<a NAME="E12:37"/>
<font color="Maroon">N</font> is    <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T44">YELLOW_9:44</a></i>;<br/>
<a NAME="E13:37"/><b>then </b>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">NN</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">NN</font> #) <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">N</font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E14:37"/>
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, , <a class="ref" href="waybel30.html#T1" target="_self">Th1</a>, <a class="ref" href="yellow12.html#T36">YELLOW12:36</a></i>;<br/>


</div>
