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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c1">T</font> <a href="compts_1.html#NR2" title="COMPTS_1:NR.2">is_T2</a> 
 ;<br/>

<b>assume </b><a NAME="E2:21"/>
 ex <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Olive" title="b1">S</font> &amp; not <font color="Olive" title="b2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">S</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:21"/><i><font color="Green" title="E19">A2</font></i>: 
 ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">x2</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E6:21"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c2">S</font>
 <b>and </b><br/><a NAME="E7:21"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c4">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c2">S</font>
 <b>and </b><br/><a NAME="E8:21"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c3">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x2</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E4:21"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">U1</font>, <font color="Maroon" title="c6">U2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E10:21"/><i><font color="Green" title="E23">A6</font></i>: 
<font color="Maroon" title="c5">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E11:21"/><i><font color="Green" title="E24">A7</font></i>: 
<font color="Maroon" title="c6">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E12:21"/><i><font color="Green" title="E25">A8</font></i>: 
<font color="Maroon" title="c3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">U1</font>
 <b>and </b><br/><a NAME="E13:21"/><i><font color="Green" title="E26">A9</font></i>: 
<font color="Maroon" title="c4">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">U2</font>
 <b>and </b><br/><a NAME="E14:21"/><i><font color="Green" title="E27">A10</font></i>: 
<font color="Maroon" title="c5">U1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">U2</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="frechet2.html#E8:21"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="compts_1.html#D4" title="COMPTS_1:def.4">COMPTS_1:def 4</a></i>;<br/>
<a NAME="E15:21"/><i><font color="Green" title="E28">A11</font></i>: 
<font color="Maroon" title="c5">U1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="frechet2.html#E14:21"><i><font color="Green" title="E27">A10</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E16:21"/>
<font color="Maroon" title="c2">S</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c3">x1</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E6:21"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="frechet.html#D4" title="FRECHET:def.4">FRECHET:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E18:21"/><i><font color="Green" title="E29">A12</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c7">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c2">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">U1</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E10:21"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="frechet2.html#E12:21"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="frechet.html#D2" title="FRECHET:def.2">FRECHET:def 2</a></i>;<br/>
<a NAME="E19:21"/>
<font color="Maroon" title="c2">S</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c4">x2</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E7:21"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="frechet.html#D4" title="FRECHET:def.4">FRECHET:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">n2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E21:21"/><i><font color="Green" title="E30">A13</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c8">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c2">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">U2</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E11:21"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="frechet2.html#E13:21"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="frechet.html#D2" title="FRECHET:def.2">FRECHET:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">n</font> =  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c8">n2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E23:21"/><i><font color="Green" title="E31">A14</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">U1</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E18:21"><i><font color="Green" title="E29">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<a NAME="E24:21"/>
<font color="Maroon" title="c2">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">U2</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E21:21"><i><font color="Green" title="E30">A13</font></i></a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<b>hence </b><a NAME="E25:21"/>
contradiction
 <b>by </b><i><a class="txt" href="frechet2.html#E15:21"><i><font color="Green" title="E28">A11</font></i></a>, <a class="txt" href="frechet2.html#E23:21"><i><font color="Green" title="E31">A14</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


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