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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="waybel_0.html#V17">infs-preserving</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E1:48"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E2:48"/>
<span class="p1"><a href="waybel_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="waybel_1.html#K1">]</a></span> is <a href="waybel_1.html#V3">Galois</a>
 
<b>by </b><i><a class="ref" href="waybel34.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:48"/><b>then </b>
 <a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">c<sub>3</sub></font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i><a class="txt" href="waybel34.html#E1:48"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel_1.html#T29">WAYBEL_1:29</a></i>;<br/>
<a NAME="E4:48"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E5:48"/><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">c<sub>5</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>5</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> #) &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>4</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>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #) )
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E6:48"/><i><font color="Green">E37</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 
;<br/>
<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>

<b>assume </b><a NAME="E7:48"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R1">=</a>  <a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E8:48"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <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>5</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>5</sub></font> #)
 
<b>by </b><i><a class="txt" href="waybel34.html#E4:48"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel34.html#E5:48"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel34.html#E6:48"><i><font color="Green">E37</font></i></a>, <a class="txt" href="waybel34.html#E7:48"><i><font color="Green">E38</font></i></a>, <a class="ref" href="waybel34.html#T31" target="_self">Th31</a></i>;<br/>
<b>thus </b><a NAME="E9:48"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> implies <font color="Maroon">c<sub>6</sub></font> is <a href="t_0topsp.html#V1">open</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 ;<br/>

<a NAME="E2:48_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> holds  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>1</sub></font></span>)</span> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="waybel34.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E3:48_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="waybel34.html#V2" target="_self">relatively_open</a>
 
<b>by </b><i><a class="txt" href="waybel34.html#E7:48"><i><font color="Green">E38</font></i></a>, <a class="ref" href="waybel34.html#T28" target="_self">Th28</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="t_0topsp.html#D2">T_0TOPSP:def 2</a><br/>

<b>assume </b><a NAME="E4:48_1"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E5:48_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel34.html#E3:48_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="waybel34.html#D9" target="_self">Def9</a></i>;<br/>
<b>hence </b><a NAME="E6:48_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="waybel34.html#E8:48"><i><font color="Green">E39</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E10:48"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="t_0topsp.html#D2">T_0TOPSP:def 2</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>7</sub></font>;<br/><a NAME="E1:48_2"/><i><font color="Green">E41</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>7</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>7</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>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #) &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>8</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> #) )
 <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>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="waybel34.html#E5:48"><i><font color="Green">E36</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="waybel34.html#E5:48"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel34.html#E1:48_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="yellow_9.html#T50">YELLOW_9:50</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="waybel34.html#E5:48"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel34.html#E1:48_2"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E5:48_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>11</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="waybel34.html#E10:48"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E6:48_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel34.html#E5:48"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel34.html#E1:48_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="yellow_9.html#T50">YELLOW_9:50</a></i>;<br/><a NAME="E7:48_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font> is <a href="waybel_0.html#V13">upper</a>
 <b>by </b><i><a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/><a NAME="E8:48_2"/><b>then </b>
(  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="waybel_0.html#T16">WAYBEL_0:16</a>, <a class="ref" href="waybel_0.html#T24">WAYBEL_0:24</a></i>;<br/><a NAME="E9:48_2"/><b>then </b>
 <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E10:48_2"/>
 <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel34.html#E7:48"><i><font color="Green">E38</font></i></a>, <a class="txt" href="waybel34.html#E1:48_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="waybel34.html#E6:48_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="waybel_0.html#T13">WAYBEL_0:13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:48"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 <b>by </b><i><a class="ref" href="waybel34.html#T23" target="_self">Th23</a></i>;<br/>


</div>
