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<span class="kw">let </span><font color="Maroon" title="c2">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font> is   <a href="lopban_1.html#NM2" title="LOPBAN_1:NM.2">RealBanachSpace</a></span><br/>

<a NAME="E1:66"/>
 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="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">seq</font> is  <a href="rsspace3.html#NV2" title="RSSPACE3:NV.2">Cauchy_sequence_by_Norm</a>  holds <br/><font color="Olive" title="b1">seq</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#T35" target="_self" onmouseover="rs('c0sp1/T35')" onmouseout="rh()">Th35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:66"/>
 <a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font> is   <a href="lopban_1.html#NM2" title="LOPBAN_1:NM.2">RealBanachSpace</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#D15" onmouseover="rs('lopban_1/D15')" onmouseout="rh()">LOPBAN_1:def 15</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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