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<b>let </b><font color="Maroon">f</font> be  <a href="asympt_0.html#V2">eventually-nonnegative</a> <a href="seq_1.html#NM1">Real_Sequence</a>, <font color="Maroon">g</font> be  <a href="asympt_0.html#V2">eventually-nonnegative</a> <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:54_1"/><i><font color="Green">E49</font></i>: 
 <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">g</font>
 ;<br/><a NAME="E2:54_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="asympt_0.html#T10">ASYMPT_0:10</a></i>;<br/><a NAME="E3:54_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="asympt_0.html#T10">ASYMPT_0:10</a></i>;<br/><a NAME="E4:54_1"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K10">Big_Omega</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="asympt_0.html#T19">ASYMPT_0:19</a></i>;<br/><b>hence </b><a NAME="E5:54_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K11">Big_Theta</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:54"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K11">Big_Theta</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:54_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">f</font>
 ;<br/><b>then consider </b><font color="Maroon">t</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:54_2_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>and </b><br/><a NAME="E4:54_2_1"/><i><font color="Green">E59</font></i>: 
 ex <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a> ex <font color="Olive">N</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">N</font> holds <br/>( <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> &amp; <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) ) )
 ;<br/><b>consider </b><font color="Maroon">c</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">N</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:54_2_1"/>
<font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E7:54_2_1"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> holds <br/>( <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> &amp; <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><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">N</font> = <font color="Maroon">N</font>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:54_2_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font>
 ;<br/><b>hence </b><a NAME="E2:54_2_1_1"/>
<font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E9:54_2_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">t</font> is <a href="asympt_0.html#V2">eventually-nonnegative</a>
 <b>by </b><i><a class="ref" href="asympt_0.html#D4">ASYMPT_0:def 4</a></i>;<br/><a NAME="E10:54_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/><a NAME="E11:54_2_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E12:54_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">g</font>
 <b>by </b><i>, , , <a class="ref" href="asympt_0.html#T12">ASYMPT_0:12</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:54_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">g</font>
 ;<br/><b>then consider </b><font color="Maroon">t</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:54_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>and </b><br/><a NAME="E5:54_2"/><i><font color="Green">E64</font></i>: 
 ex <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a> ex <font color="Olive">N</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">N</font> holds <br/>( <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> &amp; <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) ) )
 ;<br/><b>consider </b><font color="Maroon">c</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">N</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:54_2"/>
<font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E8:54_2"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> holds <br/>( <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> &amp; <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><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">N</font> = <font color="Maroon">N</font>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:54_2_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font>
 ;<br/><b>hence </b><a NAME="E2:54_2_2"/>
<font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E10:54_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">t</font> is <a href="asympt_0.html#V2">eventually-nonnegative</a>
 <b>by </b><i><a class="ref" href="asympt_0.html#D4">ASYMPT_0:def 4</a></i>;<br/><a NAME="E11:54_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/><a NAME="E12:54_2"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K10">Big_Omega</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E13:54_2"/><b>then </b>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="asympt_0.html#T19">ASYMPT_0:19</a></i>;<br/><b>hence </b><a NAME="E14:54_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">f</font>
 <b>by </b><i>, , , <a class="ref" href="asympt_0.html#T12">ASYMPT_0:12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:54"/>
 <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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