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<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="E1:22"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">T</font> <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 ;<br/>

<b>assume </b><a NAME="E2:22"/>
 ex <font color="Olive">S</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T</font> ex <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 <br/>( <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> &amp; not <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">S</font> being   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:22"/><i><font color="Green">E31</font></i>: 
 ex <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 <br/>( <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Maroon">S</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Maroon">S</font> &amp; <font color="Olive">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x2</font> )
 ;<br/>
<b>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="E6:22"/><i><font color="Green">E32</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>and </b><br/><a NAME="E7:22"/><i><font color="Green">E53</font></i>: 
<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>and </b><br/><a NAME="E8:22"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">U1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">U2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E10:22"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">U1</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E11:22"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">U2</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E12:22"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font>
 <b>and </b><br/><a NAME="E13:22"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U2</font>
 <b>and </b><br/><a NAME="E14:22"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">U1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">U2</font>
 <b>by </b><i>, , <a class="ref" href="compts_1.html#D4">COMPTS_1:def 4</a></i>;<br/>
<a NAME="E15:22"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">U1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="frechet2.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E16:22"/>
<font color="Maroon">S</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">x1</font>
 
<b>by </b><i>, <a class="ref" href="frechet.html#D4">FRECHET:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">n1</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="E18:22"/><i><font color="Green">E76</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">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E1"><i><font color="Green">Lemma24</font></i></a>, , <a class="ref" href="frechet.html#D2">FRECHET:def 2</a></i>;<br/>
<a NAME="E19:22"/>
<font color="Maroon">S</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">x2</font>
 
<b>by </b><i>, <a class="ref" href="frechet.html#D4">FRECHET:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">n2</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="E21:22"/><i><font color="Green">E77</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">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U2</font>
 <b>by </b><i>, <a class="txt" href="frechet2.html#E1:22"><i><font color="Green">E27</font></i></a>, <a class="ref" href="frechet.html#D2">FRECHET:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> =  <a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E23:22"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E24:22"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E25:22"/>
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E26:22"/><b>then </b>
<font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U2</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E4:22"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>hence </b><a NAME="E27:22"/>
contradiction
 <b>by </b><i>, <a class="txt" href="frechet2.html#E6:22"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


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