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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="lopban_1.html#NM2">RealBanachSpace</a>;<br/>



<a NAME="E1:51"/>
for <font color="Olive">seq</font> being  <a href="normsp_1.html#NM2">sequence</a> of <span class="p1">(<span class="default"><a href="rsspace4.html#K9" target="_self">R_NormSpace_of_BoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>  st <font color="Olive">seq</font> is <a href="rsspace3.html#V1">CCauchy</a> holds <br/><font color="Olive">seq</font> is <a href="normsp_1.html#V4">convergent</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:51"/>
 <a href="rsspace4.html#K9" target="_self">R_NormSpace_of_BoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> is   <a href="lopban_1.html#NM2">RealBanachSpace</a>
 <b>by </b><i><a class="ref" href="lopban_1.html#D16">LOPBAN_1:def 16</a></i>;<br/>


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