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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">Bt</font> being  <a href="yellow_8.html#NM1" title="YELLOW_8:NM.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/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#D4" target="_self" title="NORMSP_2:def.4">Def4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c4">B</font> being   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of <font color="Maroon" title="c3">x</font><span class="kw"> such that </span><br/><a NAME="E3:37_1_1_1"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c4">B</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="frechet.html#D2" title="FRECHET:def.2">FRECHET:def 2</a></span>;<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#D4" target="_self" title="NORMSP_2:def.4">Def4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">Bt</font> = <font color="Maroon" title="c4">B</font> as   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of <font color="Maroon" title="c2">xt</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#T25" target="_self" title="NORMSP_2:th.25">Th25</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c6">Bt</font> = <font color="Maroon" title="c5">Bt</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">thus </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_1_1"><span class="lab"><font color="Green" title="E25">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="frechet.html#D2" title="FRECHET:def.2">FRECHET:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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