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<div class="add">

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:23"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">T</font> is <a href="frechet.html#V1">first-countable</a>
 ;<br/>

<b>thus </b><a NAME="E2:23"/>
( <font color="Maroon">T</font> <a href="compts_1.html#NR2" target="_self">is_T2</a>  implies for <font color="Olive">S</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T</font><br/> for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Olive">S</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Olive">S</font> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font> )
 <b>by </b><i/>;<br/>

<b>assume </b><a NAME="E3:23"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">S</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T</font><br/> for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Olive">S</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Olive">S</font> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font>
 ;<br/>

<b>assume </b><a NAME="E4:23"/>
not <font color="Maroon">T</font> <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>, <font color="Maroon">x2</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:23"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font>
 <b>and </b><br/><a NAME="E7:23"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">U1</font>, <font color="Olive">U2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">U1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">U2</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Olive">U1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Olive">U2</font> holds <br/><font color="Olive">U1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">U2</font>
 <b>by </b><i><a class="ref" href="compts_1.html#D4">COMPTS_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">B1</font> being    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">x1</font><b> such that </b><br/><a NAME="E9:23"/><i><font color="Green">E64</font></i>: 
 ex <font color="Olive">S</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>(  <a href="relat_1.html#K1">dom</a> <font color="Olive">S</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> &amp; ( for <font color="Olive">n</font>, <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="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">n</font> holds <br/><font color="Olive">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> ) )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">S1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E11:23"/><i><font color="Green">E67</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E12:23"/><i><font color="Green">E70</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font>
 <b>and </b><br/><a NAME="E13:23"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">n</font>, <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="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">n</font> holds <br/><font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">B2</font> being    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">x2</font><b> such that </b><br/><a NAME="E15:23"/><i><font color="Green">E73</font></i>: 
 ex <font color="Olive">S</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>(  <a href="relat_1.html#K1">dom</a> <font color="Olive">S</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> &amp; ( for <font color="Olive">n</font>, <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="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">n</font> holds <br/><font color="Olive">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> ) )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">S2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E17:23"/><i><font color="Green">E74</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E18:23"/><i><font color="Green">E75</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font>
 <b>and </b><br/><a NAME="E19:23"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">n</font>, <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="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">n</font> holds <br/><font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E1:23"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E20:23"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:23_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<a NAME="E2:23_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</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#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">U1</font> = <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<b>reconsider </b><font color="Maroon">U1</font> = <font color="Maroon">U1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E5:23_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">U1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font> )
 
<b>by </b><i><a class="txt" href="frechet2.html#E3"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="yellow_8.html#T21">YELLOW_8:21</a></i>;<br/>
<a NAME="E6:23_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E2"><i><font color="Green">Lemma28</font></i></a>, , <a class="txt" href="frechet2.html#E6:23"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">U2</font> = <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<b>reconsider </b><font color="Maroon">U2</font> = <font color="Maroon">U2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E9:23_1"/>
( <font color="Maroon">U2</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U2</font> )
 
<b>by </b><i>, <a class="ref" href="yellow_8.html#T21">YELLOW_8:21</a></i>;<br/>
<a NAME="E10:23_1"/><b>then </b>
<font color="Maroon">U1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">U2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:23_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">U1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>;<br/>
<b>take </b>
<font color="Maroon">x</font>
;<br/>

<a NAME="E13:23_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U2</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E4"><i><font color="Green">Lemma36</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:23_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 ;<br/>

<b>thus </b><a NAME="E15:23_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="frechet2.html#E4"><i><font color="Green">Lemma36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">S</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E22:23"/><i><font color="Green">E98</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E23:23"/><i><font color="Green">E99</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E24:23"/><i><font color="Green">E100</font></i>: 
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>(<a class="txt" href="frechet2.html#E3:23"><i><font color="Green">E31</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">S</font> = <font color="Maroon">S</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <b>by </b><i><a class="txt" href="frechet2.html#E6:23"><i><font color="Green">E32</font></i></a>, <a class="txt" href="frechet2.html#E3"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E26:23"/>
<font color="Maroon">S</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">x1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">U1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="frechet.html#D2">FRECHET:def 2</a><br/>

<b>assume </b><a NAME="E1:23_2"/>
( <font color="Maroon">U1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">V1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:23_2"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">V1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 <b>and </b><br/><a NAME="E4:23_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">V1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U1</font>
 <b>by </b><i><a class="ref" href="yellow_8.html#T22">YELLOW_8:22</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:23_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">S1</font>
 <b>and </b><br/><a NAME="E7:23_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">V1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> <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/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="frechet2.html#E1"><i><font color="Green">Lemma24</font></i></a>, <a class="txt" href="frechet2.html#E5"><i><font color="Green">Lemma37</font></i></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="E9:23_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E10:23_2"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E11:23_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:23_2"/>
<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E13:23_2"/><b>then </b>
<font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:23_2"/><b>then </b>
<font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="frechet2.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E15:23_2"/>
<font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E4"><i><font color="Green">Lemma36</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:23"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="frechet.html#D4">FRECHET:def 4</a></i>;<br/>
<a NAME="E28:23"/>
<font color="Maroon">S</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3"><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">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="frechet.html#D2">FRECHET:def 2</a><br/>

<b>assume </b><a NAME="E1:23_3"/>
( <font color="Maroon">U2</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">V2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:23_3"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">V2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font>
 <b>and </b><br/><a NAME="E4:23_3"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">V2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U2</font>
 <b>by </b><i><a class="ref" href="yellow_8.html#T22">YELLOW_8:22</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:23_3"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">S2</font>
 <b>and </b><br/><a NAME="E7:23_3"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">V2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="txt" href="frechet2.html#E4"><i><font color="Green">Lemma36</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="frechet2.html#E2"><i><font color="Green">Lemma28</font></i></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="E9:23_3"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E10:23_3"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E11:23_3"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:23_3"/>
<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E13:23_3"/><b>then </b>
<font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:23_3"/><b>then </b>
<font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E15:23_3"/>
<font color="Maroon">S</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#E5"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="frechet2.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E29:23"/><b>then </b>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="frechet.html#D4">FRECHET:def 4</a></i>;<br/>
<b>hence </b><a NAME="E30:23"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div>
