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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">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>: 
for <font color="Olive" title="b1">S</font> being  <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">T</font><br/> for <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 <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> holds <br/><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>assume </b><a NAME="E2:21"/>
not <font color="Maroon" title="c1">T</font> <a href="urysohn1.html#NR1" title="URYSOHN1:NR.1">is_T1</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c4">S</font> being   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">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>: 
<font color="Maroon" title="c4">S</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finsop_1.html#K3" title="FINSOP_1:func.3">--&gt;</a> <font color="Maroon" title="c2">x1</font>
 <b>and </b><br/><a NAME="E5:21"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c2">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x2</font>
 <b>and </b><br/><a NAME="E6:21"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c4">S</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c3">x2</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a></i>;<br/>
<a NAME="E7:21"/>
<font color="Maroon" title="c4">S</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c2">x1</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E4:21"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="frechet.html#T23" title="FRECHET:th.23">FRECHET:23</a></i>;<br/>
<a NAME="E8:21"/><b>then </b><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c2">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="c4">S</font>
 
<b>by </b><i><a class="ref" href="frechet.html#D4" title="FRECHET:def.4">FRECHET:def 4</a></i>;<br/>
<a NAME="E9:21"/>
<font color="Maroon" title="c3">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="c4">S</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E6:21"><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>hence </b><a NAME="E10:21"/>
contradiction
 <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#E5:21"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="frechet2.html#E8:21"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>


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