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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="lopban_2.html#NM3" title="LOPBAN_2:NM.3">Banach_Algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">l</font>, <font color="Olive" title="b4">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>   st <font color="Olive" title="b3">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">k</font> holds <br/><span class="p1">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Olive" title="b4">k</font>,<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">l</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K6" title="LOPBAN_4:func.6">Expan_e</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">l</font></span>)</span></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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">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>   st <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> holds <br/><span class="p1">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Olive" title="b2">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K6" title="LOPBAN_4:func.6">Expan_e</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">l</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">l</font>, <font color="Maroon" title="c5">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> implies <span class="p1">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K6" title="LOPBAN_4:func.6">Expan_e</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">l</font></span>)</span> )</span><br/>









<span class="kw">assume </span><a NAME="E1:35"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon" title="c4">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K6" title="LOPBAN_4:func.6">Expan_e</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">l</font></span>)</span></span><br/>

<a NAME="E2:35"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/>
<a NAME="E3:35"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A3</font></span>: 
<font color="Maroon" title="c4">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="lab"><font color="Green" title="E37">A4</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">l</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <font color="Maroon" title="c4">l</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">l</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_4.html#D2" target="_self" title="LOPBAN_4:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <font color="Maroon" title="c4">l</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">l</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_4.html#D2" target="_self" title="LOPBAN_4:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span></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="Maroon" title="c2">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">l</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span></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="Maroon" title="c2">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">l</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T103" title="XCMPLX_1:th.103">XCMPLX_1:103</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K3" title="LOPBAN_4:func.3">Coef_e</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">l</font></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="Maroon" title="c2">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">l</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="lopban_4.html#D4" target="_self" title="LOPBAN_4:def.4">Def4</a></span>
<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" title="LOPBAN_4:func.3">Coef_e</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <font color="Maroon" title="c4">l</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="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">l</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_4.html#K6" title="LOPBAN_4:func.6">Expan_e</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">l</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="lopban_4.html#D7" target="_self" title="LOPBAN_4:def.7">Def7</a></span>
;<br/>
<a NAME="E5:35"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E38">A5</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">l</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">l</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="lopban_4.html#D8" target="_self" title="LOPBAN_4:def.8">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">l</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">l</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="bhsp_4.html#D1" title="BHSP_4:def.1">BHSP_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">l</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">l</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">l</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="lopban_4.html#D8" target="_self" title="LOPBAN_4:def.8">Def8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:35"/>
<span class="p1">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K6" title="LOPBAN_4:func.6">Expan_e</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">l</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E37">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
