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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">n</font> is   <a href="lopban_1.html#NM2" title="LOPBAN_1:NM.2">RealBanachSpace</a></span><br/>

<a NAME="E1:23"/>
for <font color="Olive" title="b1">seq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">seq</font> is <a href="rsspace3.html#V1" title="RSSPACE3:attr.1">CCauchy</a> holds <br/><font color="Olive" title="b1">seq</font> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>
 
<b>by </b><i><a class="ref" href="real_ns1.html#T12" target="_self" title="REAL_NS1:th.12">Th12</a></i>;<br/>
<b>hence </b><a NAME="E2:23"/>
 <a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">n</font> is   <a href="lopban_1.html#NM2" title="LOPBAN_1:NM.2">RealBanachSpace</a>
 <b>by </b><i><a class="ref" href="lopban_1.html#D16" title="LOPBAN_1:def.16">LOPBAN_1:def 16</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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