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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="E1:27"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<b>let </b><font color="Maroon">S1</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T1</font>;<br/><b>let </b><font color="Maroon">S2</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T2</font>;<br/>



<b>assume </b><a NAME="E2:27"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">S1</font>
 ;<br/>

<b>let </b><font color="Maroon">y</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="E3:27"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">y</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">S1</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T2</font> ;<br/>
<a NAME="E5:27"/>
<font color="Maroon">S2</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">y'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">U2</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="frechet.html#D2">FRECHET:def 2</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:27_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">U2</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:27_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U2</font>
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:27_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E5:27_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Maroon">S1</font>
 <b>and </b><br/><a NAME="E6:27_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E1"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">U1</font> = <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">U2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font> ;<br/>
<a NAME="E8:27_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">U1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, , <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<a NAME="E9:27_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">U2</font>
 
<b>by </b><i>, <a class="txt" href="frechet2.html#E1:27"><i><font color="Green">E27</font></i></a>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T1</font> ;<br/>
<a NAME="E11:27_1"/>
<font color="Maroon">S1</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="frechet.html#D4">FRECHET:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:27_1"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">U2</font>
 <b>by </b><i>, <a class="txt" href="frechet2.html#E2:27"><i><font color="Green">E31</font></i></a>, <a class="ref" href="frechet.html#D2">FRECHET:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E14:27_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E15:27_1"/><b>then </b>
<font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">U2</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E3:27"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E16:27_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">U2</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E17:27_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E18:27_1"/>
<font color="Maroon">S2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U2</font>
 <b>by </b><i>, <a class="txt" href="frechet2.html#E3"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:27"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Maroon">S2</font>
 <b>by </b><i><a class="ref" href="frechet.html#D4">FRECHET:def 4</a></i>;<br/>


</div>
