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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">xt</font> be   <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>;<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/><b>consider </b><font color="Maroon">B</font> being    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">x</font><b> such that </b><br/><a NAME="E3:38_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">B</font> is <a href="card_4.html#V1">countable</a>
 <b>by </b><i><a class="ref" href="frechet.html#D1">FRECHET:def 1</a></i>;<br/><a NAME="E4:38_1_1_1"/>
<font color="Maroon">B</font> is   <a href="struct_0.html#NM3">Subset-Family</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/><b>then reconsider </b><font color="Maroon">Bt</font> = <font color="Maroon">B</font> as    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">xt</font> <b>by </b><i/>;<br/><b>take </b><font color="Maroon">Bt</font> = <font color="Maroon">Bt</font>;<br/><b>thus </b><a NAME="E6:38_1_1_1"/>
<font color="Maroon">Bt</font> is <a href="card_4.html#V1">countable</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:38_1_1"/>
 <a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font> is <a href="frechet.html#V1">first-countable</a>
 <b>by </b><i><a class="ref" href="frechet.html#D1">FRECHET:def 1</a></i>;<br/>


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