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<div class="add">

<b>let </b><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>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:54_1"/><i><font color="Green" title="E34">A1</font></i>: 
 <a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <font color="Maroon" title="c2">g</font>
 ;<br/><a NAME="E2:54_1"/><b>then </b><i><font color="Green" title="E35">A2</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="ref" href="asympt_0.html#T10" title="ASYMPT_0:th.10">ASYMPT_0:10</a></i>;<br/><a NAME="E3:54_1"/>
<font color="Maroon" title="c2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:54_1"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="asympt_0.html#T10" title="ASYMPT_0:th.10">ASYMPT_0:10</a></i>;<br/><a NAME="E4:54_1"/><b>then </b>
<font color="Maroon" title="c1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Omega</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="ref" href="asympt_0.html#T19" title="ASYMPT_0:th.19">ASYMPT_0:19</a></i>;<br/><b>hence </b><a NAME="E5:54_1"/>
<font color="Maroon" title="c1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K11" title="ASYMPT_0:func.11">Big_Theta</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:54_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:54"/><i><font color="Green" title="E34">A3</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K11" title="ASYMPT_0:func.11">Big_Theta</a> <font color="Maroon" title="c2">g</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:54_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <font color="Maroon" title="c1">f</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c4">t</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E3:54_2_1"/><i><font color="Green" title="E35">A4</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font>
 <b>and </b><br/><a NAME="E4:54_2_1"/><i><font color="Green" title="E36">A5</font></i>: 
 ex <font color="Olive" title="b1">c</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">c</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="Maroon" title="c4">t</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b3">n</font></span>)</span> &amp; <font color="Maroon" title="c4">t</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</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/><b>consider </b><font color="Maroon" title="c5">c</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>, <font color="Maroon" title="c6">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> <b> such that </b><br/><a NAME="E6:54_2_1"/>
<font color="Maroon" title="c5">c</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> 
 <b>and </b><br/><a NAME="E7:54_2_1"/><i><font color="Green" title="E37">A6</font></i>: 
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="c6">N</font> holds <br/>( <font color="Maroon" title="c4">t</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Maroon" title="c4">t</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</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>  )
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:54_2_1"><i><font color="Green" title="E36">A5</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c7">N</font> = <font color="Maroon" title="c6">N</font>;<br/><b>let </b><font color="Maroon" title="c8">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> ;<br/><b>assume </b><a NAME="E1:54_2_1_1"/>
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">N</font>
 ;<br/><b>hence </b><a NAME="E2:54_2_1_1"/>
<font color="Maroon" title="c4">t</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c8">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> 
 <b>by </b><i><a class="txt" href="asympt_1.html#E7:54_2_1"><i><font color="Green" title="E37">A6</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E9:54_2_1"/><b>then </b><i><font color="Green" title="E38">A7</font></i>: 
<font color="Maroon" title="c4">t</font> is <a href="asympt_0.html#V2" title="ASYMPT_0:attr.2">eventually-nonnegative</a>
 <b>by </b><i><a class="ref" href="asympt_0.html#D4" title="ASYMPT_0:def.4">ASYMPT_0:def 4</a></i>;<br/><a NAME="E10:54_2_1"/><i><font color="Green" title="E39">A8</font></i>: 
<font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:54_2_1"><i><font color="Green" title="E36">A5</font></i></a></i>;<br/><a NAME="E11:54_2_1"/>
<font color="Maroon" title="c1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:54"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E12:54_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:54_2_1"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="asympt_1.html#E9:54_2_1"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="asympt_1.html#E10:54_2_1"><i><font color="Green" title="E39">A8</font></i></a>, <a class="ref" href="asympt_0.html#T12" title="ASYMPT_0:th.12">ASYMPT_0:12</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:54_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <font color="Maroon" title="c2">g</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c4">t</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E4:54_2"/><i><font color="Green" title="E35">A9</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font>
 <b>and </b><br/><a NAME="E5:54_2"/><i><font color="Green" title="E36">A10</font></i>: 
 ex <font color="Olive" title="b1">c</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">c</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="Maroon" title="c4">t</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b3">n</font></span>)</span> &amp; <font color="Maroon" title="c4">t</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</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/><b>consider </b><font color="Maroon" title="c5">c</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>, <font color="Maroon" title="c6">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> <b> such that </b><br/><a NAME="E7:54_2"/>
<font color="Maroon" title="c5">c</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> 
 <b>and </b><br/><a NAME="E8:54_2"/><i><font color="Green" title="E37">A11</font></i>: 
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="c6">N</font> holds <br/>( <font color="Maroon" title="c4">t</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Maroon" title="c4">t</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</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>  )
 <b>by </b><i><a class="txt" href="asympt_1.html#E5:54_2"><i><font color="Green" title="E36">A10</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c7">N</font> = <font color="Maroon" title="c6">N</font>;<br/><b>let </b><font color="Maroon" title="c8">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> ;<br/><b>assume </b><a NAME="E1:54_2_2"/>
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">N</font>
 ;<br/><b>hence </b><a NAME="E2:54_2_2"/>
<font color="Maroon" title="c4">t</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c8">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> 
 <b>by </b><i><a class="txt" href="asympt_1.html#E8:54_2"><i><font color="Green" title="E37">A11</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E10:54_2"/><b>then </b><i><font color="Green" title="E38">A12</font></i>: 
<font color="Maroon" title="c4">t</font> is <a href="asympt_0.html#V2" title="ASYMPT_0:attr.2">eventually-nonnegative</a>
 <b>by </b><i><a class="ref" href="asympt_0.html#D4" title="ASYMPT_0:def.4">ASYMPT_0:def 4</a></i>;<br/><a NAME="E11:54_2"/><i><font color="Green" title="E39">A13</font></i>: 
<font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E5:54_2"><i><font color="Green" title="E36">A10</font></i></a></i>;<br/><a NAME="E12:54_2"/>
<font color="Maroon" title="c1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Omega</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:54"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E13:54_2"/><b>then </b>
<font color="Maroon" title="c2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="ref" href="asympt_0.html#T19" title="ASYMPT_0:th.19">ASYMPT_0:19</a></i>;<br/><b>hence </b><a NAME="E14:54_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:54_2"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="asympt_1.html#E10:54_2"><i><font color="Green" title="E38">A12</font></i></a>, <a class="txt" href="asympt_1.html#E11:54_2"><i><font color="Green" title="E39">A13</font></i></a>, <a class="ref" href="asympt_0.html#T12" title="ASYMPT_0:th.12">ASYMPT_0:12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:54"/>
 <a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div>
