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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="lopban_2.html#NM3">Banach_Algebra</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>









<b>assume </b><a NAME="E1:35"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:35"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E3:35"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="lopban_4.html#E1:35"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:35"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<b>then </b><i><font color="Green">E36</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="lopban_4.html#E1:35"><i><font color="Green">E34</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/>
<b>then </b><i><font color="Green">E37</font></i>: <span class="p1">(<span class="default"><a href="lopban_4.html#K7" target="_self">Alfa</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lopban_4.html#E3:35"><i><font color="Green">E35</font></i></a>, <a class="ref" href="lopban_4.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lopban_4.html#E5:35"><i><font color="Green">E36</font></i></a>, <a class="ref" href="lopban_3.html#D1">LOPBAN_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43">LOPBAN_3:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K7" target="_self">Alfa</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lopban_4.html#E1:35"><i><font color="Green">E34</font></i></a>, <a class="ref" href="lopban_4.html#D8" target="_self">Def8</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lopban_3.html#K11">#N</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_4.html#D2" target="_self">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lopban_3.html#K11">#N</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_3.html#K11">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_4.html#D2" target="_self">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lopban_3.html#K11">#N</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_3.html#K11">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43">LOPBAN_3:43</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lopban_3.html#K11">#N</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_3.html#K11">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T103">XCMPLX_1:103</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K3" target="_self">Coef_e</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lopban_3.html#K11">#N</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_3.html#K11">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lopban_4.html#E3:35"><i><font color="Green">E35</font></i></a>, <a class="ref" href="lopban_4.html#D4" target="_self">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K3" target="_self">Coef_e</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lopban_3.html#K11">#N</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_3.html#K11">#N</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43">LOPBAN_3:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_4.html#K6" target="_self">Expan_e</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="lopban_4.html#E3:35"><i><font color="Green">E35</font></i></a>, <a class="ref" href="lopban_4.html#D7" target="_self">Def7</a></i>
;<br/>
<b>hence </b><a NAME="E8:35"/>
<span class="p1">(<span class="default"><a href="lopban_4.html#K7" target="_self">Alfa</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K7" target="_self">Alfa</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K6" target="_self">Expan_e</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="lopban_4.html#E6:35"><i><font color="Green">E37</font></i></a></i>;<br/>


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