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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">xt</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">Bt</font> being   <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c2">xt</font> st <font color="Olive" title="b1">Bt</font> is <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a></span><br/><b>reconsider </b><font color="Maroon" title="c3">x</font> = <font color="Maroon" title="c2">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/><b>consider </b><font color="Maroon" title="c4">B</font> being    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c3">x</font><b> such that </b><br/><a NAME="E3:37_1_1_1"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c4">B</font> is <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>
 <b>by </b><i><a class="ref" href="frechet.html#D1" title="FRECHET:def.1">FRECHET:def 1</a></i>;<br/><a NAME="E4:37_1_1_1"/>
<font color="Maroon" title="c4">B</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</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/><b>then reconsider </b><font color="Maroon" title="c5">Bt</font> = <font color="Maroon" title="c4">B</font> as    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c2">xt</font> <b>by </b><i><a class="ref" href="normsp_2.html#T25" target="_self" title="NORMSP_2:th.25">Th25</a></i>;<br/><b>take </b><font color="Maroon" title="c6">Bt</font> = <font color="Maroon" title="c5">Bt</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">Bt</font> is <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a></span><br/><b>thus </b><a NAME="E6:37_1_1_1"/>
<font color="Maroon" title="c6">Bt</font> is <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>
 <b>by </b><i><a class="txt" href="#E3:37_1_1_1"><i><font color="Green" title="E25">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:37_1_1"/>
 <a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font> is <a href="frechet.html#V1" title="FRECHET:attr.1">first-countable</a>
 <b>by </b><i><a class="ref" href="frechet.html#D1" title="FRECHET:def.1">FRECHET:def 1</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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