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

<b>let </b><font color="Maroon">c<sub>1</sub></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">E20</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">c<sub>1</sub></font></span>)</span> is <a href="waybel_3.html#V3">continuous</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">c<sub>1</sub></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">E21</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_2.html#K4">inf_op</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="yellow_3.html#K3">:]</a></span>;<br/>
<a NAME="E4:37"/><i><font color="Green">E22</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></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">c<sub>4</sub></font> =  <a href="waybel_2.html#K4">inf_op</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E6:37"/><i><font color="Green">E23</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> #)
 
;<br/>
<a NAME="E7:37"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></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">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="yellow_3.html#K3">:]</a></span> <b>by </b><i><a class="txt" href="waybel30.html#E4:37"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E2:37_1"/>
 <a href="waybel_2.html#K4">inf_op</a> <font color="Maroon">c<sub>1</sub></font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="waybel30.html#E4:37"><i><font color="Green">E22</font></i></a>, <a class="txt" href="waybel30.html#E6:37"><i><font color="Green">E23</font></i></a>, <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">E25</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is    <a href="waybel21.html#M1">SemilatticeHomomorphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel21.html#T5">WAYBEL21:5</a></i>;<br/>
<a NAME="E9:37"/>
<font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></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">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="yellow_3.html#K3">:]</a></span> <b>by </b><i><a class="txt" href="waybel30.html#E4:37"><i><font color="Green">E22</font></i></a>, <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">c<sub>1</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="waybel30.html#E4:37"><i><font color="Green">E22</font></i></a>, <a class="txt" href="waybel30.html#E6:37"><i><font color="Green">E23</font></i></a>, <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">E26</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel30.html#E7:37"><i><font color="Green">E24</font></i></a>, <a class="txt" href="waybel30.html#E8:37"><i><font color="Green">E25</font></i></a>, <a class="ref" href="waybel21.html#T46">WAYBEL21:46</a></i>;<br/>
<a NAME="E11:37"/><i><font color="Green">E27</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> #)
 
;<br/>
<a NAME="E12:37"/>
<font color="Maroon">c<sub>1</sub></font> is    <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>3</sub></font> #) <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="txt" href="waybel30.html#E1:37"><i><font color="Green">E20</font></i></a>, <a class="ref" href="waybel30.html#T20" target="_self">Th20</a></i>;<br/>
<b>hence </b><a NAME="E14:37"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="waybel30.html#E2:37"><i><font color="Green">E21</font></i></a>, <a class="txt" href="waybel30.html#E10:37"><i><font color="Green">E26</font></i></a>, <a class="txt" href="waybel30.html#E11:37"><i><font color="Green">E27</font></i></a>, <a class="ref" href="yellow12.html#T36">YELLOW12:36</a></i>;<br/>


</div>
