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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="clopban2.html#NM3" title="CLOPBAN2:NM.3">Complex_Banach_Algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">z</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K3" title="CLOPBAN4:func.3">Expan_e</a> (<font color="Olive" title="b3">k</font>,<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">w</font>)</span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b3">k</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">z</font>, <font color="Maroon" title="c3">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K3" title="CLOPBAN4:func.3">Expan_e</a> (<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">k</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K3" title="CLOPBAN4:func.3">Expan_e</a> (<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span><br/>







<a NAME="E1:31"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">k</font>
 
;<br/>
<a NAME="E2:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A1</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="clopban4.html#K3" title="CLOPBAN4:func.3">Expan_e</a> (<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c4">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K7" title="SIN_COS:func.7">Coef_e</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban4.html#D3" target="_self" title="CLOPBAN4:def.3">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K7" title="SIN_COS:func.7">Coef_e</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c4">k</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="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T39" title="CLOPBAN3:th.39">CLOPBAN3:39</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K7" title="SIN_COS:func.7">Coef_e</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c4">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T38" title="CLOPBAN3:th.38">CLOPBAN3:38</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a></span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c4">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="complex1.html#D4" title="COMPLEX1:def.4">COMPLEX1:def 4</a>, <a class="ref" href="sin_cos.html#T1" title="SIN_COS:th.1">SIN_COS:1</a>, <a class="ref" href="sin_cos.html#D7" title="SIN_COS:def.7">SIN_COS:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c4">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban4.html#D1" target="_self" title="CLOPBAN4:def.1">Def2</a>, <a class="ref" href="complex1.html#D4" title="COMPLEX1:def.4">COMPLEX1:def 4</a></span>
;<br/> <a class="txt" 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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