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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">c</font>, <font color="Maroon" title="c2">c1</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="partfun3.html#V1" title="PARTFUN3:attr.1">positive-yielding</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Maroon" title="c2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c1">c</font> holds <br/> for <font color="Olive" title="b2">x</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><a href="asympt_3.html#NK1" title="ASYMPT_3:NK.1">polynom</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_3.html#NK1" title="ASYMPT_3:NK.1">polynom</a> <font color="Maroon" title="c1">c</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c1">c</font> implies  for <font color="Olive" title="b1">x</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><a href="asympt_3.html#NK1" title="ASYMPT_3:NK.1">polynom</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_3.html#NK1" title="ASYMPT_3:NK.1">polynom</a> <font color="Maroon" title="c1">c</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">x</font></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:13"/><span class="lab"><font color="Green" title="E8">AS</font></span>: 
<font color="Maroon" title="c2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c1">c</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><a href="asympt_3.html#NK1" title="ASYMPT_3:NK.1">polynom</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_3.html#NK1" title="ASYMPT_3:NK.1">polynom</a> <font color="Maroon" title="c1">c</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="asympt_3.html#NK1" title="ASYMPT_3:NK.1">polynom</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_3.html#NK1" title="ASYMPT_3:NK.1">polynom</a> <font color="Maroon" title="c1">c</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span><br/>

<span class="lab"><font color="Green" title="E9">D1</font></span>:  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c1</font> <a href="asympt_2.html#K1" title="ASYMPT_2:func.1">(#)</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> (<font color="Maroon" title="c4">x</font>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span></span>)</span> = 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c2">c1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_2.html#T26" onmouseover="rs('asympt_2/T26')" onmouseout="rh()">ASYMPT_2:26</a></span>
<br/>.= 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c1">c</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" onmouseover="rs('valued_1/D5')" onmouseout="rh()">VALUED_1:def 5</a>, <a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E8">AS</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E10">D2</font></span>:  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="asympt_2.html#K1" title="ASYMPT_2:func.1">(#)</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> (<font color="Maroon" title="c4">x</font>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span></span>)</span></span>)</span> = 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="asympt_2.html#K1" title="ASYMPT_2:func.1">(#)</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> (<font color="Maroon" title="c4">x</font>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" onmouseover="rs('valued_1/D5')" onmouseout="rh()">VALUED_1:def 5</a></span>
<br/>.= 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c1">c</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_2.html#T26" onmouseover="rs('asympt_2/T26')" onmouseout="rh()">ASYMPT_2:26</a></span>
;<br/>
<a NAME="E4:13"/>
 for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c1</font> <a href="asympt_2.html#K1" title="ASYMPT_2:func.1">(#)</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> (<font color="Maroon" title="c4">x</font>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">c1</font> <a href="asympt_2.html#K1" title="ASYMPT_2:func.1">(#)</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> (<font color="Maroon" title="c4">x</font>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="asympt_2.html#K1" title="ASYMPT_2:func.1">(#)</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> (<font color="Maroon" title="c4">x</font>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c1</font> <a href="asympt_2.html#K1" title="ASYMPT_2:func.1">(#)</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> (<font color="Maroon" title="c4">x</font>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">c1</font> <a href="asympt_2.html#K1" title="ASYMPT_2:func.1">(#)</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> (<font color="Maroon" title="c4">x</font>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="asympt_2.html#K1" title="ASYMPT_2:func.1">(#)</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> (<font color="Maroon" title="c4">x</font>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:13_3"/><span class="lab"><font color="Green" title="E11">D3</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c1</font> <a href="asympt_2.html#K1" title="ASYMPT_2:func.1">(#)</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> (<font color="Maroon" title="c4">x</font>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">c1</font> <a href="asympt_2.html#K1" title="ASYMPT_2:func.1">(#)</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> (<font color="Maroon" title="c4">x</font>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="asympt_2.html#K1" title="ASYMPT_2:func.1">(#)</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> (<font color="Maroon" title="c4">x</font>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span><br/>

<a NAME="E2:13_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">D4</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c2">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_2.html#T26" onmouseover="rs('asympt_2/T26')" onmouseout="rh()">ASYMPT_2:26</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">c1</font> <a href="asympt_2.html#K1" title="ASYMPT_2:func.1">(#)</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> (<font color="Maroon" title="c4">x</font>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> (<font color="Maroon" title="c4">x</font>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_3"><span class="lab"><font color="Green" title="E12">D4</font></span></a>, <a class="ref" href="asympt_2.html#T26" onmouseover="rs('asympt_2/T26')" onmouseout="rh()">ASYMPT_2:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> (<font color="Maroon" title="c4">x</font>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E8">AS</font></span></a>, <a class="ref" href="valued_1.html#T6" onmouseover="rs('valued_1/T6')" onmouseout="rh()">VALUED_1:6</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> (<font color="Maroon" title="c4">x</font>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="asympt_2.html#K1" title="ASYMPT_2:func.1">(#)</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> (<font color="Maroon" title="c4">x</font>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_3"><span class="lab"><font color="Green" title="E11">D3</font></span></a>, <a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E9">D1</font></span></a>, <a class="ref" href="asympt_2.html#T26" onmouseover="rs('asympt_2/T26')" onmouseout="rh()">ASYMPT_2:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="asympt_2.html#K1" title="ASYMPT_2:func.1">(#)</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> (<font color="Maroon" title="c4">x</font>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T6" onmouseover="rs('valued_1/T6')" onmouseout="rh()">VALUED_1:6</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">P2</font></span>: 
<font color="Maroon" title="c2">c1</font> <a href="asympt_2.html#K1" title="ASYMPT_2:func.1">(#)</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> (<font color="Maroon" title="c4">x</font>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="asympt_2.html#K1" title="ASYMPT_2:func.1">(#)</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> (<font color="Maroon" title="c4">x</font>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E9">D1</font></span></a>, <a class="txt" href="#E3:13"><span class="lab"><font color="Green" title="E10">D2</font></span></a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="asympt_3.html#NK1" title="ASYMPT_3:NK.1">polynom</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">x</font> = 
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c1</font> <a href="asympt_2.html#K1" title="ASYMPT_2:func.1">(#)</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> (<font color="Maroon" title="c4">x</font>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_2.html#D2" onmouseover="rs('asympt_2/D2')" onmouseout="rh()">ASYMPT_2:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="asympt_2.html#K1" title="ASYMPT_2:func.1">(#)</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> (<font color="Maroon" title="c4">x</font>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T64" onmouseover="rs('afinsq_2/T64')" onmouseout="rh()">AFINSQ_2:64</a>, <a class="txt" href="#E5:13"><span class="lab"><font color="Green" title="E11">P2</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_3.html#NK1" title="ASYMPT_3:NK.1">polynom</a> <font color="Maroon" title="c1">c</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_2.html#D2" onmouseover="rs('asympt_2/D2')" onmouseout="rh()">ASYMPT_2:def 2</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
