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<span class="kw">let </span><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">g</font> be   <a href="asympt_0.html#V2" title="ASYMPT_0:attr.2">eventually-nonnegative</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="asympt_0.html#K11" title="ASYMPT_0:func.11">Big_Omega</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K5" title="ASYMPT_0:func.5">+</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_0.html#K11" title="ASYMPT_0:func.11">Big_Omega</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">N1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E2:49"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">N1</font> holds <br/><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_0.html#D2" target="_self" title="ASYMPT_0:def.2">Def4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">N2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:49"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">N2</font> holds <br/><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_0.html#D2" target="_self" title="ASYMPT_0:def.2">Def4</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K11" title="ASYMPT_0:func.11">Big_Omega</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K5" title="ASYMPT_0:func.5">+</a> <font color="Maroon" title="c2">g</font></span>)</span> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K11" title="ASYMPT_0:func.11">Big_Omega</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span> ) &amp; ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K11" title="ASYMPT_0:func.11">Big_Omega</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K11" title="ASYMPT_0:func.11">Big_Omega</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K5" title="ASYMPT_0:func.5">+</a> <font color="Maroon" title="c2">g</font></span>)</span> ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K11" title="ASYMPT_0:func.11">Big_Omega</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K11" title="ASYMPT_0:func.11">Big_Omega</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K5" title="ASYMPT_0:func.5">+</a> <font color="Maroon" title="c2">g</font></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:49_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K11" title="ASYMPT_0:func.11">Big_Omega</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K5" title="ASYMPT_0:func.5">+</a> <font color="Maroon" title="c2">g</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K11" title="ASYMPT_0:func.11">Big_Omega</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">t</font> being    <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#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)<span class="kw"> such that </span><br/><a NAME="E3:49_1_1"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">and </span><br/><a NAME="E4:49_1_1"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
 ex <font color="Olive" title="b1">d</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ex <font color="Olive" title="b2">N</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; (  for <font color="Olive" title="b3">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">N</font> holds <br/>( <font color="Olive" title="b1">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K5" title="ASYMPT_0:func.5">+</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">n</font> &amp; <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) )
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c7">d</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>, <font color="Maroon" title="c8">N</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E6:49_1_1"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E7:49_1_1"/><span class="lab"><font color="Green" title="E33">A6</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">N</font> holds <br/>( <font color="Maroon" title="c7">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K5" title="ASYMPT_0:func.5">+</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49_1_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">a</font> =  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c8">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c3">N1</font>,<font color="Maroon" title="c4">N2</font>)</span>)</span>);<br/><a NAME="E8:49_1_1"/><span class="lab"><font color="Green" title="E34">A7</font></span>: 
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c8">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c3">N1</font>,<font color="Maroon" title="c4">N2</font>)</span>)</span>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><a NAME="E9:49_1_1"/><span class="lab"><font color="Green" title="E35">A8</font></span>: 
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c8">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c3">N1</font>,<font color="Maroon" title="c4">N2</font>)</span>)</span>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c3">N1</font>,<font color="Maroon" title="c4">N2</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><a NAME="E10:49_1_1"/>
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c3">N1</font>,<font color="Maroon" title="c4">N2</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">N2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><a NAME="E11:49_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A9</font></span>: 
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c8">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c3">N1</font>,<font color="Maroon" title="c4">N2</font>)</span>)</span>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">N2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:49_1_1"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:49_1_1"/>
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c3">N1</font>,<font color="Maroon" title="c4">N2</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">N1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><a NAME="E13:49_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A10</font></span>: 
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c8">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c3">N1</font>,<font color="Maroon" title="c4">N2</font>)</span>)</span>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">N1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:49_1_1"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c8">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c3">N1</font>,<font color="Maroon" title="c4">N2</font>)</span>)</span>) implies ( <font color="Maroon" title="c7">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> &amp; <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/><span class="kw">assume </span><a NAME="E1:49_1_1_1"/><span class="lab"><font color="Green" title="E38">A11</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c8">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c3">N1</font>,<font color="Maroon" title="c4">N2</font>)</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> &amp; <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><a NAME="E2:49_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">N1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:49_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:49_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A12</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E28">A1</font></span></a></span>;<br/><a NAME="E4:49_1_1_1"/>
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">N2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:49_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E1:49_1_1_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E5:49_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A13</font></span>: 
<font color="Maroon" title="c2">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49"><span class="lab"><font color="Green" title="E29">A2</font></span></a></span>;<br/><a NAME="E6:49_1_1_1"/><span class="lab"><font color="Green" title="E41">A14</font></span>: 
<span class="p1">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_0.html#D7" target="_self" title="ASYMPT_0:def.7">Def10</a></span>;<br/><a NAME="E7:49_1_1_1"/>
<span class="p1">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K5" title="ASYMPT_0:func.5">+</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:49_1_1_1_1_1"/>
( <span class="p1">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> or <span class="p1">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49_1_1_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:49_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K5" title="ASYMPT_0:func.5">+</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span><br/><div class="add"><a NAME="E2:49_1_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49_1_1_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:49_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K5" title="ASYMPT_0:func.5">+</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T7" title="SEQ_1:th.7">SEQ_1:7</a></span>; <a class="txt" 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="49_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:49_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K5" title="ASYMPT_0:func.5">+</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span><br/><div class="add"><a NAME="E2:49_1_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_1_1_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:49_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K5" title="ASYMPT_0:func.5">+</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T7" title="SEQ_1:th.7">SEQ_1:7</a></span>; <a class="txt" 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="E8:49_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A15</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K5" title="ASYMPT_0:func.5">+</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49_1_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E9:49_1_1_1"/><span class="lab"><font color="Green" title="E43">A16</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:49_1_1"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E1:49_1_1_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:49_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K5" title="ASYMPT_0:func.5">+</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49_1_1"><span class="lab"><font color="Green" title="E33">A6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E11:49_1_1_1"/>
<font color="Maroon" title="c7">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:49_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><span class="kw">thus </span><a NAME="E12:49_1_1_1"/>
<font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49_1_1"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E9:49_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a></span>; <a class="txt" 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="E15:49_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K11" title="ASYMPT_0:func.11">Big_Omega</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_1_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E6:49_1_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a></span>; <a class="txt" 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">assume </span><a NAME="E2:49_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K11" title="ASYMPT_0:func.11">Big_Omega</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K11" title="ASYMPT_0:func.11">Big_Omega</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K5" title="ASYMPT_0:func.5">+</a> <font color="Maroon" title="c2">g</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">t</font> being    <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#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)<span class="kw"> such that </span><br/><a NAME="E4:49_1"/><span class="lab"><font color="Green" title="E30">A17</font></span>: 
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">and </span><br/><a NAME="E5:49_1"/><span class="lab"><font color="Green" title="E31">A18</font></span>: 
 ex <font color="Olive" title="b1">d</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ex <font color="Olive" title="b2">N</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; (  for <font color="Olive" title="b3">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">N</font> holds <br/>( <font color="Olive" title="b1">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">n</font> &amp; <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) )
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c7">d</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>, <font color="Maroon" title="c8">N</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:49_1"/><span class="lab"><font color="Green" title="E32">A19</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E8:49_1"/><span class="lab"><font color="Green" title="E33">A20</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">N</font> holds <br/>( <font color="Maroon" title="c7">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49_1"><span class="lab"><font color="Green" title="E31">A18</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">a</font> =  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c8">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c3">N1</font>,<font color="Maroon" title="c4">N2</font>)</span>)</span>);<br/><a NAME="E9:49_1"/><span class="lab"><font color="Green" title="E34">A21</font></span>: 
<font color="Maroon" title="c8">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c8">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c3">N1</font>,<font color="Maroon" title="c4">N2</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><div><span class="lab"><font color="Green" title="E35">A22</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c8">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c3">N1</font>,<font color="Maroon" title="c4">N2</font>)</span>)</span>) implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K5" title="ASYMPT_0:func.5">+</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> &amp; <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/><a NAME="E1:49_1_2"/>
( <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>) &amp; <font color="Maroon" title="c2">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><a NAME="E2:49_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><a NAME="E3:49_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E4:49_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K5" title="ASYMPT_0:func.5">+</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T7" title="SEQ_1:th.7">SEQ_1:7</a></span>;<br/><a NAME="E5:49_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K5" title="ASYMPT_0:func.5">+</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K2" title="ASYMPT_0:func.2">max</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49_1"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E6:49_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A23</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K5" title="ASYMPT_0:func.5">+</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_0.html#D7" target="_self" title="ASYMPT_0:def.7">Def10</a></span>;<br/><span class="kw">assume </span><a NAME="E7:49_1_2"/>
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c8">N</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c3">N1</font>,<font color="Maroon" title="c4">N2</font>)</span>)</span>)
 ; <a class="txt" 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="c7">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K5" title="ASYMPT_0:func.5">+</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> &amp; <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><a NAME="E8:49_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A24</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:49_1"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:49_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:49_1"><span class="lab"><font color="Green" title="E33">A20</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:49_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K5" title="ASYMPT_0:func.5">+</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49_1_2"><span class="lab"><font color="Green" title="E36">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><span class="kw">thus </span><a NAME="E11:49_1_2"/>
<font color="Maroon" title="c6">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:49_1"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="txt" href="#E8:49_1_2"><span class="lab"><font color="Green" title="E37">A24</font></span></a></span>; <a class="txt" 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="E11:49_1"/>
<font color="Maroon" title="c7">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49_1"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><span class="kw">hence </span><a NAME="E12:49_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K11" title="ASYMPT_0:func.11">Big_Omega</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K5" title="ASYMPT_0:func.5">+</a> <font color="Maroon" title="c2">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49_1"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="txt" href="#E10:49_1"><span class="lab"><font color="Green" title="E35">A22</font></span></a></span>; <a class="txt" 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="E6:49"/>
 <a href="asympt_0.html#K11" title="ASYMPT_0:func.11">Big_Omega</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K5" title="ASYMPT_0:func.5">+</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_0.html#K11" title="ASYMPT_0:func.11">Big_Omega</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K9" title="ASYMPT_0:func.9">max</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
