<?xml version="1.0"?>
<div class="add">

<span class="kw">set </span><font color="Maroon" title="c2">X</font> =  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">m</font> = <span class="p1">(<span class="default"><a href="funcsdom.html#K6" title="FUNCSDOM:func.6">RealFuncMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">a</font> = <span class="p1">(<span class="default"><a href="funcsdom.html#K5" title="FUNCSDOM:func.5">RealFuncAdd</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">E</font> = <span class="p1">(<span class="default"><a href="funcsdom.html#K7" title="FUNCSDOM:func.7">RealFuncExtMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">O</font> =  <a href="funcsdom.html#K9" title="FUNCSDOM:func.9">RealFuncUnit</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">Z</font> =  <a href="funcsdom.html#K8" title="FUNCSDOM:func.8">RealFuncZero</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>;<br/>
<a NAME="E1:22_1_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>,<a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><span class="p1">(<span class="default"><a href="funcsdom.html#K6" title="FUNCSDOM:func.6">RealFuncMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>,<a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies <span class="p1">(<span class="default"><a href="funcsdom.html#K6" title="FUNCSDOM:func.6">RealFuncMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_1_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>,<a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.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"> <span class="p1">(<span class="default"><a href="funcsdom.html#K6" title="FUNCSDOM:func.6">RealFuncMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c10">g</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:22_1_1_1"/><span class="lab"><font color="Green" title="E11">B2</font></span>: 
( <font color="Maroon" title="c9">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>  &amp; <font color="Maroon" title="c10">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>  &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">f</font>,<font color="Maroon" title="c10">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c12">g</font> = <font color="Maroon" title="c10">g</font> as   <a href="asympt_3.html#V1" title="ASYMPT_3:attr.1">polynomially-abs-bounded</a>  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#D2" target="_self" onmouseover="rs('asympt_3/D2')" onmouseout="rh()">DefX1</a>, <a class="txt" href="#E3:22_1_1_1"><span class="lab"><font color="Green" title="E11">B2</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">h</font> = <span class="p1">(<span class="default"><a href="funcsdom.html#K6" title="FUNCSDOM:func.6">RealFuncMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">f1</font> = <font color="Maroon" title="c11">f</font>, <font color="Maroon" title="c15">g1</font> = <font color="Maroon" title="c12">g</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" onmouseover="rs('funct_2/T8')" onmouseout="rh()">FUNCT_2:8</a></span>;<br/>
<a NAME="E6:22_1_1_1"/><span class="lab"><font color="Green" title="E12">B5</font></span>: 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6" title="FUNCSDOM:func.6">RealFuncMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K6" title="FUNCSDOM:func.6">RealFuncMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="funcsdom.html#K1" title="FUNCSDOM:func.1">.</a> (<font color="Maroon" title="c14">f1</font>,<font color="Maroon" title="c15">g1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_1_1_1"><span class="lab"><font color="Green" title="E11">B2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">h</font> = <span class="p1">(<span class="default"><a href="funcsdom.html#K6" title="FUNCSDOM:func.6">RealFuncMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_1_1_1"><span class="lab"><font color="Green" title="E12">B5</font></span></a></span>;<br/>
<a NAME="E8:22_1_1_1"/>
<font color="Maroon" title="c16">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c11">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c12">g</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funct_2.html#D8" onmouseover="rs('funct_2/D8')" onmouseout="rh()">FUNCT_2:def 8</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="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c12">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c16">h</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_1_1_1"><span class="lab"><font color="Green" title="E12">B5</font></span></a>, <a class="ref" href="funcsdom.html#T2" onmouseover="rs('funcsdom/T2')" onmouseout="rh()">FUNCSDOM:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c12">g</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T5" onmouseover="rs('valued_1/T5')" onmouseout="rh()">VALUED_1:5</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>
<span class="kw">hence </span><a NAME="E9:22_1_1_1"/>
<span class="p1">(<span class="default"><a href="funcsdom.html#K6" title="FUNCSDOM:func.6">RealFuncMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#D2" target="_self" onmouseover="rs('asympt_3/D2')" onmouseout="rh()">DefX1</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="E2:22_1_1"/><span class="kw">then </span>
 <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>  is   <a href="realset1.html#NM1" title="REALSET1:NM.1">Preserv</a> of  <a href="funcsdom.html#K6" title="FUNCSDOM:func.6">RealFuncMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="realset1.html#D1" onmouseover="rs('realset1/D1')" onmouseout="rh()">REALSET1:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">m</font> = <span class="p1">(<span class="default"><a href="funcsdom.html#K6" title="FUNCSDOM:func.6">RealFuncMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="realset1.html#T2" onmouseover="rs('realset1/T2')" onmouseout="rh()">REALSET1:2</a></span>;<br/>
<a NAME="E4:22_1_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>,<a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><span class="p1">(<span class="default"><a href="funcsdom.html#K5" title="FUNCSDOM:func.5">RealFuncAdd</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>,<a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies <span class="p1">(<span class="default"><a href="funcsdom.html#K5" title="FUNCSDOM:func.5">RealFuncAdd</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_1_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>,<a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.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"> <span class="p1">(<span class="default"><a href="funcsdom.html#K5" title="FUNCSDOM:func.5">RealFuncAdd</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">f</font>, <font color="Maroon" title="c11">g</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:22_1_1_2"/><span class="lab"><font color="Green" title="E11">B2</font></span>: 
( <font color="Maroon" title="c10">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>  &amp; <font color="Maroon" title="c11">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>  &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">f</font> = <font color="Maroon" title="c10">f</font>, <font color="Maroon" title="c13">g</font> = <font color="Maroon" title="c11">g</font> as   <a href="asympt_3.html#V1" title="ASYMPT_3:attr.1">polynomially-abs-bounded</a>  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#D2" target="_self" onmouseover="rs('asympt_3/D2')" onmouseout="rh()">DefX1</a>, <a class="txt" href="#E3:22_1_1_2"><span class="lab"><font color="Green" title="E11">B2</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">h</font> = <span class="p1">(<span class="default"><a href="funcsdom.html#K5" title="FUNCSDOM:func.5">RealFuncAdd</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">f1</font> = <font color="Maroon" title="c12">f</font>, <font color="Maroon" title="c16">g1</font> = <font color="Maroon" title="c13">g</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" onmouseover="rs('funct_2/T8')" onmouseout="rh()">FUNCT_2:8</a></span>;<br/>
<a NAME="E6:22_1_1_2"/><span class="lab"><font color="Green" title="E12">B5</font></span>: 
<span class="p1">(<span class="default"><a href="funcsdom.html#K5" title="FUNCSDOM:func.5">RealFuncAdd</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K5" title="FUNCSDOM:func.5">RealFuncAdd</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="funcsdom.html#K1" title="FUNCSDOM:func.1">.</a> (<font color="Maroon" title="c15">f1</font>,<font color="Maroon" title="c16">g1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_1_1_2"><span class="lab"><font color="Green" title="E11">B2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">h</font> = <span class="p1">(<span class="default"><a href="funcsdom.html#K5" title="FUNCSDOM:func.5">RealFuncAdd</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>
<a NAME="E8:22_1_1_2"/>
<font color="Maroon" title="c17">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c12">f</font> <a href="series_1.html#K1" title="SERIES_1:func.1">+</a> <font color="Maroon" title="c13">g</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funct_2.html#D8" onmouseover="rs('funct_2/D8')" onmouseout="rh()">FUNCT_2:def 8</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="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="series_1.html#K1" title="SERIES_1:func.1">+</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c17">h</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c18">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c18">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_1_1_2"><span class="lab"><font color="Green" title="E12">B5</font></span></a>, <a class="ref" href="funcsdom.html#T1" onmouseover="rs('funcsdom/T1')" onmouseout="rh()">FUNCSDOM:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="series_1.html#K1" title="SERIES_1:func.1">+</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c18">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T1" onmouseover="rs('valued_1/T1')" onmouseout="rh()">VALUED_1:1</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>
<span class="kw">hence </span><a NAME="E9:22_1_1_2"/>
<span class="p1">(<span class="default"><a href="funcsdom.html#K5" title="FUNCSDOM:func.5">RealFuncAdd</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#D2" target="_self" onmouseover="rs('asympt_3/D2')" onmouseout="rh()">DefX1</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="E5:22_1_1"/><span class="kw">then </span>
 <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>  is   <a href="realset1.html#NM1" title="REALSET1:NM.1">Preserv</a> of  <a href="funcsdom.html#K5" title="FUNCSDOM:func.5">RealFuncAdd</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="realset1.html#D1" onmouseover="rs('realset1/D1')" onmouseout="rh()">REALSET1:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">a</font> = <span class="p1">(<span class="default"><a href="funcsdom.html#K5" title="FUNCSDOM:func.5">RealFuncAdd</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="realset1.html#T2" onmouseover="rs('realset1/T2')" onmouseout="rh()">REALSET1:2</a></span>;<br/>
<a NAME="E7:22_1_1"/><span class="lab"><font color="Green" title="E11">Q0</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T95" onmouseover="rs('zfmisc_1/T95')" onmouseout="rh()">ZFMISC_1:95</a></span>;<br/>
<a NAME="E8:22_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K7" title="FUNCSDOM:func.7">RealFuncExtMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E9:22_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">Q1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K7" title="FUNCSDOM:func.7">RealFuncExtMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</a>, <a class="txt" href="#E7:22_1_1"><span class="lab"><font color="Green" title="E11">Q0</font></span></a></span>;<br/>
<a NAME="E10:22_1_1"/>
 for <font color="Olive" title="b1">h</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K7" title="FUNCSDOM:func.7">RealFuncExtMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> holds <br/><font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">h</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="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K7" title="FUNCSDOM:func.7">RealFuncExtMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> implies <font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_1_1_3"/>
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K7" title="FUNCSDOM:func.7">RealFuncExtMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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"> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:22_1_1_3"/><span class="lab"><font color="Green" title="E13">Q2</font></span>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K7" title="FUNCSDOM:func.7">RealFuncExtMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> &amp; <font color="Maroon" title="c10">h</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="funcsdom.html#K7" title="FUNCSDOM:func.7">RealFuncExtMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">s</font>, <font color="Maroon" title="c13">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:22_1_1_3"/><span class="lab"><font color="Green" title="E14">B2</font></span>: 
( <font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c13">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>  )
 <span class="kw">and </span><br/><a NAME="E6:22_1_1_3"/><span class="lab"><font color="Green" title="E15">B3</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">s</font>,<font color="Maroon" title="c13">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_1_1"><span class="lab"><font color="Green" title="E12">Q1</font></span></a>, <a class="txt" href="#E3:22_1_1_3"><span class="lab"><font color="Green" title="E13">Q2</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">f</font> = <font color="Maroon" title="c13">f</font> as   <a href="asympt_3.html#V1" title="ASYMPT_3:attr.1">polynomially-abs-bounded</a>  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#D2" target="_self" onmouseover="rs('asympt_3/D2')" onmouseout="rh()">DefX1</a>, <a class="txt" href="#E5:22_1_1_3"><span class="lab"><font color="Green" title="E14">B2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">s</font> = <font color="Maroon" title="c12">s</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_1_1_3"><span class="lab"><font color="Green" title="E14">B2</font></span></a></span>;<br/>
<a NAME="E9:22_1_1_3"/><span class="lab"><font color="Green" title="E16">H1</font></span>: 
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K7" title="FUNCSDOM:func.7">RealFuncExtMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_1_1"><span class="lab"><font color="Green" title="E12">Q1</font></span></a>, <a class="txt" href="#E3:22_1_1_3"><span class="lab"><font color="Green" title="E13">Q2</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">f1</font> = <font color="Maroon" title="c14">f</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" onmouseover="rs('funct_2/T8')" onmouseout="rh()">FUNCT_2:8</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">h1</font> = <font color="Maroon" title="c10">h</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_1_1_3"><span class="lab"><font color="Green" title="E16">H1</font></span></a>, <a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</a>, <a class="txt" href="#E3:22_1_1_3"><span class="lab"><font color="Green" title="E13">Q2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">h1</font> = <font color="Maroon" title="c17">h1</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>
<a NAME="E13:22_1_1_3"/>
<font color="Maroon" title="c18">h1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c15">s</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c14">f</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funct_2.html#D8" onmouseover="rs('funct_2/D8')" onmouseout="rh()">FUNCT_2:def 8</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="c18">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">s</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c14">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c18">h1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c19">n</font> = 
<font color="Maroon" title="c15">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">f1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c19">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_1_1_3"><span class="lab"><font color="Green" title="E15">B3</font></span></a>, <a class="txt" href="#E9:22_1_1_3"><span class="lab"><font color="Green" title="E16">H1</font></span></a>, <a class="ref" href="funcsdom.html#T4" onmouseover="rs('funcsdom/T4')" onmouseout="rh()">FUNCSDOM:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">s</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c14">f</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c19">n</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>
<span class="kw">hence </span><a NAME="E14:22_1_1_3"/>
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#D2" target="_self" onmouseover="rs('asympt_3/D2')" onmouseout="rh()">DefX1</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="E11:22_1_1"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K7" title="FUNCSDOM:func.7">RealFuncExtMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">E</font> = <span class="p1">(<span class="default"><a href="funcsdom.html#K7" title="FUNCSDOM:func.7">RealFuncExtMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_1_1"><span class="lab"><font color="Green" title="E12">Q1</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">O</font> =  <a href="funcsdom.html#K9" title="FUNCSDOM:func.9">RealFuncUnit</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ;<br/>
<a NAME="E14:22_1_1"/>
( <font color="Maroon" title="c11">O</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) &amp;  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c11">O</font> is  <a href="asympt_3.html#V1" title="ASYMPT_3:attr.1">polynomially-abs-bounded</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T10" target="_self" onmouseover="rs('asympt_3/T10')" onmouseout="rh()">LM1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">O</font> = <font color="Maroon" title="c11">O</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#D2" target="_self" onmouseover="rs('asympt_3/D2')" onmouseout="rh()">DefX1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">Z</font> =  <a href="funcsdom.html#K8" title="FUNCSDOM:func.8">RealFuncZero</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ;<br/>
<a NAME="E17:22_1_1"/>
( <font color="Maroon" title="c13">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) &amp;  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c13">Z</font> is  <a href="asympt_3.html#V1" title="ASYMPT_3:attr.1">polynomially-abs-bounded</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T10" target="_self" onmouseover="rs('asympt_3/T10')" onmouseout="rh()">LM1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">Z</font> = <font color="Maroon" title="c13">Z</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#D2" target="_self" onmouseover="rs('asympt_3/D2')" onmouseout="rh()">DefX1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">S</font> =  <a href="funcsdom.html#G1" title="FUNCSDOM:aggr.1">AlgebraStr</a>(# <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>,<font color="Maroon" title="c8">m</font>,<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">E</font>,<font color="Maroon" title="c12">O</font>,<font color="Maroon" title="c14">Z</font> #);<br/>
<a NAME="E19:22_1_1"/>
 <a href="funcsdom.html#G1" title="FUNCSDOM:aggr.1">AlgebraStr</a>(# <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>,<font color="Maroon" title="c8">m</font>,<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">E</font>,<font color="Maroon" title="c12">O</font>,<font color="Maroon" title="c14">Z</font> #) is   <a href="funcsdom.html#V1" title="FUNCSDOM:attr.1">strict</a>   <a href="funcsdom.html#L1" title="FUNCSDOM:struct.1">AlgebraStr</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E20:22_1_1"/>
(  ex <font color="Olive">b<sub>1</sub></font> being   <a href="funcsdom.html#V1" title="FUNCSDOM:attr.1">strict</a>   <a href="funcsdom.html#L1" title="FUNCSDOM:struct.1">AlgebraStr</a>  st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>  &amp;  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K6" title="FUNCSDOM:func.6">RealFuncMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a> &amp;  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K5" title="FUNCSDOM:func.5">RealFuncAdd</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a> &amp;  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K7" title="FUNCSDOM:func.7">RealFuncExtMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcsdom.html#K9" title="FUNCSDOM:func.9">RealFuncUnit</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> &amp;  the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcsdom.html#K8" title="FUNCSDOM:func.8">RealFuncZero</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ) &amp; (  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="funcsdom.html#V1" title="FUNCSDOM:attr.1">strict</a>   <a href="funcsdom.html#L1" title="FUNCSDOM:struct.1">AlgebraStr</a>   st  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>  &amp;  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K6" title="FUNCSDOM:func.6">RealFuncMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a> &amp;  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K5" title="FUNCSDOM:func.5">RealFuncAdd</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a> &amp;  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K7" title="FUNCSDOM:func.7">RealFuncExtMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcsdom.html#K9" title="FUNCSDOM:func.9">RealFuncUnit</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> &amp;  the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcsdom.html#K8" title="FUNCSDOM:func.8">RealFuncZero</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a>  &amp;  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K6" title="FUNCSDOM:func.6">RealFuncMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a> &amp;  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K5" title="FUNCSDOM:func.5">RealFuncAdd</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a> &amp;  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K7" title="FUNCSDOM:func.7">RealFuncExtMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcsdom.html#K9" title="FUNCSDOM:func.9">RealFuncUnit</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> &amp;  the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcsdom.html#K8" title="FUNCSDOM:func.8">RealFuncZero</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></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"> verum</span><br/>


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