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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>;<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 <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/><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#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E28">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/>

<div><i><font color="Green" title="E29">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1"/>
<font color="Maroon" title="c3">St</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a>
 ;<br/><b>then consider </b><font color="Maroon" title="c4">xt</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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><b> such that </b><br/><a NAME="E3:40_1"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c3">St</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c4">xt</font>
 <b>by </b><i><a class="ref" href="frechet.html#D3" title="FRECHET:def.3">FRECHET:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">xt</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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/><a NAME="E5:40_1"/>
<font color="Maroon" title="c2">S</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="normsp_2.html#E1:40"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="normsp_2.html#T26" target="_self" title="NORMSP_2:th.26">Th26</a>, <a class="txt" href="normsp_2.html#E3:40_1"><i><font color="Green" title="E30">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E6:40_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/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_2"/>
<font color="Maroon" title="c2">S</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a>
 ;<br/><b>then consider </b><font color="Maroon" title="c4">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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> such that </b><br/><a NAME="E3:40_2"/><i><font color="Green" title="E30">A4</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="c4">x</font>
 <b>by </b><i><a class="ref" href="frechet.html#D3" title="FRECHET:def.3">FRECHET:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">xt</font> = <font color="Maroon" title="c4">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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> <b>by </b><i><a class="ref" href="normsp_2.html#D4" target="_self" title="NORMSP_2:def.4">Def4</a></i>;<br/><a NAME="E5:40_2"/>
<font color="Maroon" title="c3">St</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c5">xt</font>
 <b>by </b><i><a class="txt" href="normsp_2.html#E1:40"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="normsp_2.html#T26" target="_self" title="NORMSP_2:th.26">Th26</a>, <a class="txt" href="normsp_2.html#E3:40_2"><i><font color="Green" title="E30">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E6:40_2"/>
<font color="Maroon" title="c3">St</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/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:40"/>
( <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="frechet.html#V2" title="FRECHET:attr.2">convergent</a> )
 <b>by </b><i><a class="txt" href="normsp_2.html#E2:40"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/>


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