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<div><span class="kw">theorem </span><a NAME="T41"><span class="comment"><font color="firebrick">:: LOPBAN_4:41</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">s</font>, <font color="Olive" title="b4">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds <br/> ( <span class="p1">(<span class="default"><a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">z</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">z</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b4">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">z</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">z</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">t</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">z</font></span>)</span> &amp;  <a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b4">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">t</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">z</font></span>)</span> &amp;  <a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">z</font></span>)</span>, <a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">z</font></span>)</span> <a href="lopban_4.html#R1" title="LOPBAN_4:pred.1">are_commutative</a>  )</div></div>
