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<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>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/>







<b>assume </b><a NAME="E1:45"/><i><font color="Green">E32</font></i>: 
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>
 ;<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="E2:45"/><i><font color="Green">E33</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/>

<b>let </b><font color="Maroon">c<sub>7</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>4</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel34.html#D9" target="_self">WAYBEL34:def 9</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <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>7</sub></font></span>)</span> as  <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="txt" href="waybel34.html#E1:45"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E4:45"/>
<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#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K9">/\</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>
 
<b>by </b><i><a class="txt" href="waybel34.html#E2:45"><i><font color="Green">E33</font></i></a>, <a class="ref" href="waybel34.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E5:45"/><b>then </b><i><font color="Green">E34</font></i>: 
( <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#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</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="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E6:45"/><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> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <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="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font> as   <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> ;<br/>
<a NAME="E8:45"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</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#E5:45"><i><font color="Green">E34</font></i></a>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<b>hence </b><a NAME="E9:45"/>
<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="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


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