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<span class="kw">let </span><font color="Maroon" title="c2">c</font>, <font color="Maroon" title="c3">s</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">  ex <font color="Olive" title="b1">d</font> being   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>  ex <font color="Olive" title="b2">N</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b3">x</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">x</font> holds <br/><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="c2">c</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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="c3">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_3.html#NK1" title="ASYMPT_3:NK.1">polynom</a> <font color="Olive" title="b1">d</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">x</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">a1</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>, <font color="Maroon" title="c5">k1</font>, <font color="Maroon" title="c6">N1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E2:62"/><span class="lab"><font color="Green" title="E34">P1</font></span>: 
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">a1</font> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k1</font> &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c6">N1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> holds <br/><span class="p1">(<span class="default"><a href="asympt_3.html#NK1" title="ASYMPT_3:NK.1">polynom</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c5">k1</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T35" target="_self" onmouseover="rs('asympt_3/T35')" onmouseout="rh()">LRNG1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">a2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>, <font color="Maroon" title="c8">k2</font>, <font color="Maroon" title="c9">N2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:62"/><span class="lab"><font color="Green" title="E35">P2</font></span>: 
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">a2</font> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k2</font> &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c9">N2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> holds <br/><span class="p1">(<span class="default"><a href="asympt_3.html#NK1" title="ASYMPT_3:NK.1">polynom</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">a2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c8">k2</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T35" target="_self" onmouseover="rs('asympt_3/T35')" onmouseout="rh()">LRNG1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">d</font> being   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> <span class="kw"> such that </span><br/><a NAME="E6:62"/><span class="lab"><font color="Green" title="E36">P3</font></span>: 
 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"><font color="Maroon" title="c4">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">a2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">k2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_3.html#NK1" title="ASYMPT_3:NK.1">polynom</a> <font color="Maroon" title="c10">d</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T36" target="_self" onmouseover="rs('asympt_3/T36')" onmouseout="rh()">LRNG2</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">N</font> = <font color="Maroon" title="c6">N1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">N2</font>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">d</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">  ex <font color="Olive" title="b1">N</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b2">x</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">x</font> holds <br/><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="c2">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> <a href="real_1.html#K4" title="REAL_1:func.4">*</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="c3">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_3.html#NK1" title="ASYMPT_3:NK.1">polynom</a> <font color="Maroon" title="c10">d</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">x</font></span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c6">N1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">N2</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>  st <font color="Maroon" title="c6">N1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">N2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> holds <br/><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="c2">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> <a href="real_1.html#K4" title="REAL_1:func.4">*</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="c3">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_3.html#NK1" title="ASYMPT_3:NK.1">polynom</a> <font color="Maroon" title="c10">d</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">x</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c12">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"> ( <font color="Maroon" title="c6">N1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">N2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">x</font> implies <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="c2">c</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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="c3">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_3.html#NK1" title="ASYMPT_3:NK.1">polynom</a> <font color="Maroon" title="c10">d</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E7:62"/><span class="lab"><font color="Green" title="E37">P4</font></span>: 
<font color="Maroon" title="c6">N1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">N2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">x</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"> <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="c2">c</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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="c3">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_3.html#NK1" title="ASYMPT_3:NK.1">polynom</a> <font color="Maroon" title="c10">d</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">x</font></span><br/>

<a NAME="E8:62"/>
<font color="Maroon" title="c6">N1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">N1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">N2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a></span>;<br/>
<a NAME="E9:62"/><span class="kw">then </span>
<font color="Maroon" title="c6">N1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a>, <a class="txt" href="#E7:62"><span class="lab"><font color="Green" title="E37">P4</font></span></a></span>;<br/>
<a NAME="E10:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">P5</font></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">c</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c5">k1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E34">P1</font></span></a></span>;<br/>
<a NAME="E11:62"/>
<font color="Maroon" title="c9">N2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">N1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">N2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a></span>;<br/>
<a NAME="E12:62"/><span class="kw">then </span>
<font color="Maroon" title="c9">N2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a>, <a class="txt" href="#E7:62"><span class="lab"><font color="Green" title="E37">P4</font></span></a></span>;<br/>
<a NAME="E13:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">P6</font></span>: 
<span class="p1">(<span class="default"><a href="asympt_3.html#NK1" title="ASYMPT_3:NK.1">polynom</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">a2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c8">k2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62"><span class="lab"><font color="Green" title="E35">P2</font></span></a></span>;<br/>
<a NAME="E14:62"/><span class="lab"><font color="Green" title="E40">P7</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="asympt_3.html#NK1" title="ASYMPT_3:NK.1">polynom</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T8" target="_self" onmouseover="rs('asympt_3/T8')" onmouseout="rh()">NLM3</a></span>;<br/>
<a NAME="E15:62"/><span class="lab"><font color="Green" title="E41">P8</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="asympt_3.html#NK1" title="ASYMPT_3:NK.1">polynom</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T8" target="_self" onmouseover="rs('asympt_3/T8')" onmouseout="rh()">NLM3</a></span>;<br/>
<a NAME="E16:62"/><span class="lab"><font color="Green" title="E42">P9</font></span>: 
<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="c2">c</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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="c3">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c5">k1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c8">k2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T66" onmouseover="rs('xreal_1/T66')" onmouseout="rh()">XREAL_1:66</a>, <a class="txt" href="#E10:62"><span class="lab"><font color="Green" title="E38">P5</font></span></a>, <a class="txt" href="#E13:62"><span class="lab"><font color="Green" title="E39">P6</font></span></a>, <a class="txt" href="#E14:62"><span class="lab"><font color="Green" title="E40">P7</font></span></a>, <a class="txt" href="#E15:62"><span class="lab"><font color="Green" title="E41">P8</font></span></a></span>;<br/>
<a NAME="E17:62"/><span class="lab"><font color="Green" title="E43">P10</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c5">k1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c8">k2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">a2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">k2</font></span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:62_1_1"/>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c12">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_1"/><span class="lab"><font color="Green" title="E44">P11</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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"><font color="Maroon" title="c4">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c5">k1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c8">k2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">a2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">k2</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:62_1_1_1"/><span class="lab"><font color="Green" title="E45">P12</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c5">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E34">P1</font></span></a>, <a class="txt" href="#E1:62_1_1_1"><span class="lab"><font color="Green" title="E44">P11</font></span></a>, <a class="ref" href="power.html#D2" onmouseover="rs('power/D2')" onmouseout="rh()">POWER:def 2</a></span>;<br/><a NAME="E3:62_1_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">k2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62"><span class="lab"><font color="Green" title="E35">P2</font></span></a>, <a class="txt" href="#E1:62_1_1_1"><span class="lab"><font color="Green" title="E44">P11</font></span></a>, <a class="ref" href="power.html#D2" onmouseover="rs('power/D2')" onmouseout="rh()">POWER:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:62_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c5">k1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c8">k2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">a2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">k2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_1_1_1"><span class="lab"><font color="Green" title="E45">P12</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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"><font color="Maroon" title="c4">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c5">k1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c8">k2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">a2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">k2</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:62_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c5">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c8">k2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">k2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T27" onmouseover="rs('power/T27')" onmouseout="rh()">POWER:27</a></span>;<br/><span class="kw">hence </span><a NAME="E3:62_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c5">k1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c8">k2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">a2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">k2</font></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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E18:62"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">a2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">k2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_3.html#NK1" title="ASYMPT_3:NK.1">polynom</a> <font color="Maroon" title="c10">d</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E36">P3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:62"/>
<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="c2">c</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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="c3">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_3.html#NK1" title="ASYMPT_3:NK.1">polynom</a> <font color="Maroon" title="c10">d</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:62"><span class="lab"><font color="Green" title="E42">P9</font></span></a>, <a class="txt" href="#E17:62"><span class="lab"><font color="Green" title="E43">P10</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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>
