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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <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  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</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="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>









<b>assume </b><a NAME="E1:53"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V5">continuous</a> )
 ;<br/>

<a NAME="E2:53"/><i><font color="Green">E49</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
;<br/>
<a NAME="E3:53"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> 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>5</sub></font> <a href="relset_1.html#K11">"</a> <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:53_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:53_1"/><b>then </b>
<font color="Maroon">c<sub>6</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>2</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:53_1"/><i><font color="Green">E50</font></i>: 
( <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>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</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>2</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E6:53_1"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E4:53_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E7:53_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:53"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<a NAME="E8:53_1"/><i><font color="Green">E52</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E9:53_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:53"><i><font color="Green">E48</font></i></a>, <a class="ref" href="relat_1.html#T168">RELAT_1:168</a></i>;<br/>
<a NAME="E10:53_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K9">/\</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>2</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E11:53_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:53"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jgraph_1.html#E4:53_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="relat_1.html#T178">RELAT_1:178</a></i>;<br/>
<a NAME="E12:53_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E2:53"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jgraph_1.html#E8:53_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E13:53_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E14:53_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E4:53_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jgraph_1.html#E8:53_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="relat_1.html#T178">RELAT_1:178</a></i>;<br/>
<b>hence </b><a NAME="E15:53_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="jgraph_1.html#E7:53_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jgraph_1.html#E9:53_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jgraph_1.html#E11:53_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:53"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>


</div>
