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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E11">Th9</font></span>: <a NAME="T9"><span class="comment"><font color="firebrick">:: LOPBAN_4:9</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">X</font> being   <a href="lopban_2.html#NM3" title="LOPBAN_2:NM.3">Banach_Algebra</a><br/>  for <font color="Olive" title="b2">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">X</font><br/>  for <font color="Olive" title="b3">seq</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Olive" title="b1">X</font> holds <br/> (  <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">z</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">*</a> <font color="Olive" title="b3">seq</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">z</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">*</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Olive" title="b3">seq</font></span>)</span> &amp;  <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">seq</font> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Olive" title="b2">z</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Olive" title="b3">seq</font></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Olive" title="b2">z</font> )</div></div>
