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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E29">Th19</font></span>: <a NAME="T19"><span class="comment"><font color="firebrick">:: LOPBAN_4:19</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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b3">z</font>, <font color="Olive" title="b4">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">X</font>  st <font color="Olive" title="b3">z</font>,<font color="Olive" title="b4">w</font> <a href="lopban_4.html#R1" title="LOPBAN_4:pred.1">are_commutative</a>  holds <br/><span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">n</font> <a href="newton.html#K2" title="NEWTON:func.2">!</a></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b4">w</font></span>)</span> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K6" title="LOPBAN_4:func.6">Expan_e</a> (<font color="Olive" title="b2">n</font>,<font color="Olive" title="b3">z</font>,<font color="Olive" title="b4">w</font>)</span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">n</font></div></div>
