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<b>let </b><font color="Maroon">X</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">S</font> be   <a href="normsp_1.html#NM2">sequence</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">St</font> be   <a href="normsp_1.html#NM2">sequence</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:40"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">St</font>
 ;<br/>

<div><i><font color="Green">E41</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">St</font> is <a href="frechet.html#V2">convergent</a>
 ;<br/><b>then consider </b><font color="Maroon">xt</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E3:40_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">St</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">xt</font>
 <b>by </b><i><a class="ref" href="frechet.html#D3">FRECHET:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">xt</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/><a NAME="E5:40_1"/>
<font color="Maroon">S</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E6:40_1"/>
<font color="Maroon">S</font> is <a href="frechet.html#V2">convergent</a>
 <b>by </b><i><a class="ref" href="frechet.html#D3">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">S</font> is <a href="frechet.html#V2">convergent</a>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E3:40_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">S</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="frechet.html#D3">FRECHET:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">xt</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/><a NAME="E5:40_2"/>
<font color="Maroon">St</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">xt</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E6:40_2"/>
<font color="Maroon">St</font> is <a href="frechet.html#V2">convergent</a>
 <b>by </b><i><a class="ref" href="frechet.html#D3">FRECHET:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:40"/>
( <font color="Maroon">St</font> is <a href="frechet.html#V2">convergent</a> iff <font color="Maroon">S</font> is <a href="frechet.html#V2">convergent</a> )
 <b>by </b><i/>;<br/>


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