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

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">St</font> be   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:43"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">St</font>
 ;<br/>

<a NAME="E2:43"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="normsp_2.html#D4" target="_self" title="NORMSP_2:def.4">Def4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">SSt</font> = <font color="Maroon" title="c3">St</font> as   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span> ;<br/>
<a NAME="E4:43"/>
( <font color="Maroon" title="c3">St</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a> iff <font color="Maroon" title="c4">SSt</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a> )
 
<b>by </b><i><a class="ref" href="normsp_2.html#T27" target="_self" title="NORMSP_2:th.27">Th27</a></i>;<br/>
<b>hence </b><a NAME="E5:43"/>
( <font color="Maroon" title="c3">St</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a> iff <font color="Maroon" title="c2">S</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> )
 <b>by </b><i><a class="ref" href="normsp_2.html#T13" target="_self" title="NORMSP_2:th.13">Th13</a>, <a class="txt" href="normsp_2.html#E1:43"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/>


</div>
