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<b>let </b><font color="Maroon">X</font> be   <a href="clopban2.html#NM3">Complex_Banach_Algebra</a>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>







<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><a href="clopban4.html#K4" target="_self">Expan_e</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="clvect_1.html#D14">CLVECT_1:def 14</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:27_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a> 
 <b>by </b><i><a class="ref" href="sin_cos.html#T1">SIN_COS:1</a></i>;<br/><b>thus </b><i><font color="Green">E55</font></i>: <span class="p1">(<span class="default"><a href="clopban4.html#K4" target="_self">Expan_e</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">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#K9">Coef_e</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="sin_cos.html#D11">SIN_COS:def 11</a></i>
;<br/><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <a href="complex1.html#K6">1r</a> </span>)</span> <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><a href="clopban4.html#K4" target="_self">Expan_e</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K13">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K13">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K13">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="sin_cos.html#D10">SIN_COS:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="clvect_1.html#D14">CLVECT_1:def 14</a></i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:27_1"/>
<span class="p1">(<span class="default"><a href="clopban4.html#K4" target="_self">Expan_e</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:27"/>
 <a href="clopban4.html#K4" target="_self">Expan_e</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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