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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="E1:41"/><i><font color="Green" title="E35">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>let </b><font color="Maroon" title="c2">S</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>thus </b><a NAME="E2:41"/>
( <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">x</font> implies ( <font color="Maroon" title="c2">S</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="frechet2.html#K4" title="FRECHET2:func.4">lim</a> <font color="Maroon" title="c2">S</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1"/><i><font color="Green" title="E36">A2</font></i>: 
<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">x</font>
 ;<br/>

<b>hence </b><a NAME="E2:41_1"/>
<font color="Maroon" title="c2">S</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a>
 <b>by </b><i><a class="ref" href="frechet.html#D3" title="FRECHET:def.3">FRECHET:def 3</a></i>;<br/>

<a NAME="E3:41_1"/><b>then </b>
 ex <font color="Olive" title="b1">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="frechet2.html#E1:41"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="frechet2.html#T27" target="_self" title="FRECHET2:th.27">Th27</a></i>;<br/>
<b>hence </b><a NAME="E4:41_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="frechet2.html#K4" title="FRECHET2:func.4">lim</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E1:41_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="frechet2.html#D3" target="_self" title="FRECHET2:def.3">Def3</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:41"/><i><font color="Green" title="E36">A3</font></i>: 
( <font color="Maroon" title="c2">S</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="frechet2.html#K4" title="FRECHET2:func.4">lim</a> <font color="Maroon" title="c2">S</font> )
 ;<br/>

<a NAME="E4:41"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="frechet2.html#E1:41"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="frechet2.html#T27" target="_self" title="FRECHET2:th.27">Th27</a></i>;<br/>
<b>hence </b><a NAME="E5:41"/>
<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">x</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E3:41"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="frechet2.html#D3" target="_self" title="FRECHET2:def.3">Def3</a></i>;<br/>


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