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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="E1:14_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> &amp; <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> )
 ;<br/><a NAME="E2:14_1"/><b>then </b>
(  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">g</font> &amp;  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</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#T11">ASYMPT_0:11</a></i>;<br/><b>hence </b><a NAME="E3:14_1"/>
 <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="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:14"/>
(  <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> implies ( <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> &amp; <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/>


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