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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <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">X</font><br/> for <font color="Olive" title="b2">St</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">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>  st <font color="Olive" title="b1">S</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b2">St</font> holds <br/>( <font color="Olive" title="b2">St</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a> iff <font color="Olive" title="b1">S</font> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> )</span><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">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">St</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">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>  st <font color="Maroon" title="c2">S</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">St</font> holds <br/>( <font color="Olive" title="b1">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#V3" title="NORMSP_1:attr.3">convergent</a> )</span><br/><b>let </b><font color="Maroon" title="c3">St</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> implies ( <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#V3" title="NORMSP_1:attr.3">convergent</a> ) )</span><br/>





<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E15">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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#V3" title="NORMSP_1:attr.3">convergent</a> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c4">Sm</font> = <font color="Maroon" title="c3">St</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K2" title="NORMSP_2:func.2">MetricSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span> ;<br/>
<a NAME="E3:20"/>
( <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">Sm</font> is <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a> )
 
<b>by </b><i><a class="ref" href="frechet2.html#T32" title="FRECHET2:th.32">FRECHET2:32</a></i>;<br/>
<b>hence </b><a NAME="E4:20"/>
( <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#V3" title="NORMSP_1:attr.3">convergent</a> )
 <b>by </b><i><a class="txt" href="#E1:20"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="normsp_2.html#T5" target="_self" title="NORMSP_2:th.5">Th5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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