<?xml version="1.0"?>
<div class="add">

<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#L1" title="PRE_TOPC:struct.1">TopStruct</a> ;<br/>

<b>thus </b><a NAME="E1:37"/>
( <font color="Maroon" title="c1">T</font> is <a href="frechet.html#V3" title="FRECHET:attr.3">Frechet</a> implies for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> holds   <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Olive" title="b1">A</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1"/><i><font color="Green" title="E31">A1</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="frechet.html#V3" title="FRECHET:attr.3">Frechet</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c3">A'</font> = <font color="Maroon" title="c2">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<b>thus </b><a NAME="E3:37_1"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_1_1"/><i><font color="Green" title="E32">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c4">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E3:37_1_1"/>
 ex <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> st <br/>(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A'</font> &amp; <font color="Maroon" title="c5">x'</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> )
 
<b>by </b><i><a class="txt" href="frechet2.html#E1:37_1"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="frechet2.html#E1:37_1_1"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="frechet.html#D5" title="FRECHET:def.5">FRECHET:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:37_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="frechet2.html#D2" target="_self" title="FRECHET2:def.2">Def2</a></i>;<br/>


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

<b>thus </b><a NAME="E4:37_1"/>
 <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="frechet2.html#T17" target="_self" title="FRECHET2:th.17">Th17</a></i>;<br/>


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

<b>assume </b><a NAME="E2:37"/><i><font color="Green" title="E31">A3</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> holds   <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Olive" title="b1">A</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="frechet.html#D5" title="FRECHET:def.5">FRECHET:def 5</a><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>assume </b><a NAME="E3:37"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<a NAME="E4:37"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E2:37"><i><font color="Green" title="E31">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:37"/>
 ex <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> st <br/>(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c3">x</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> )
 <b>by </b><i><a class="ref" href="frechet2.html#D2" target="_self" title="FRECHET2:def.2">Def2</a></i>;<br/>


</div>
