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<b>let </b><font color="Maroon">T1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">T2</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">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T1</font>,<font color="Maroon">T2</font>;<br/>



<b>assume </b><a NAME="E1:28"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">T1</font> is <a href="frechet.html#V4">sequential</a>
 ;<br/>

<b>thus </b><a NAME="E2:28"/>
( <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a> implies for <font color="Olive">S1</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T1</font><br/> for <font color="Olive">S2</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T2</font>  st <font color="Olive">S2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Olive">S1</font> holds <br/><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="frechet.html#K1">Lim</a> <font color="Olive">S1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="frechet.html#K1">Lim</a> <font color="Olive">S2</font> )
 <b>by </b><i/>;<br/>

<b>assume </b><a NAME="E3:28"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">S1</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T1</font><br/> for <font color="Olive">S2</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T2</font>  st <font color="Olive">S2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Olive">S1</font> holds <br/><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="frechet.html#K1">Lim</a> <font color="Olive">S1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="frechet.html#K1">Lim</a> <font color="Olive">S2</font>
 ;<br/>

<b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a><br/>

<b>assume </b><a NAME="E4:28"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font> ;<br/>
<a NAME="E6:28"/>
for <font color="Olive">S</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T1</font>  st <font color="Olive">S</font> is <a href="frechet.html#V2">convergent</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> holds <br/> <a href="frechet.html#K1">Lim</a> <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">S</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T1</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:28_1"/>
<font color="Maroon">S</font> is <a href="frechet.html#V2">convergent</a>
 <b>and </b><br/><a NAME="E2:28_1"/><i><font color="Green">E53</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:28_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Maroon">S</font>
 ;<br/>

<b>set </b><font color="Maroon">S2</font> = <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">S</font>;<br/>
<a NAME="E4:28_1"/><i><font color="Green">E67</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:28_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="frechet.html#K1">Lim</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E6:28_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="frechet.html#K1">Lim</a> <font color="Maroon">S</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="frechet.html#K1">Lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:28_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i><a class="txt" href="frechet2.html#E2"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">B'</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font> ;<br/>
<a NAME="E9:28_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">S</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">S</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:28_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">S</font></span>)</span>
 <b>and </b><br/><a NAME="E4:28_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:28_1_1"/><i><font color="Green">E75</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="normsp_1.html#T17">NORMSP_1:17</a></i>;<br/>
<a NAME="E6:28_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E7:28_1_1"/><b>then </b>
<font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E4:28"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:28_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E9:28_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font>
 <b>by </b><i>, <a class="txt" href="frechet2.html#E3:28"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:28_1"/><b>then </b>
 <a href="frechet.html#K1">Lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">S</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B'</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E1"><i><font color="Green">Lemma24</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E11:28_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="txt" href="frechet2.html#E1:28"><i><font color="Green">E27</font></i></a>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:28"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, <a class="ref" href="frechet.html#D6">FRECHET:def 6</a></i>;<br/>


</div>
