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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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K12" title="ASYMPT_0:func.12">Big_Theta</a> <font color="Maroon" title="c2">g</font> implies <font color="Maroon" title="c2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K12" title="ASYMPT_0:func.12">Big_Theta</a> <font color="Maroon" title="c1">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:53"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K12" title="ASYMPT_0:func.12">Big_Theta</a> <font color="Maroon" title="c2">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K12" title="ASYMPT_0:func.12">Big_Theta</a> <font color="Maroon" title="c1">f</font></span><br/>

<a NAME="E2:53"/><span class="kw">then </span>
<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_Omega</a> <font color="Maroon" title="c2">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:53"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_0.html#T19" target="_self" title="ASYMPT_0:th.19">Th19</a></span>;<br/>
<a NAME="E4:53"/>
<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_Oh</a> <font color="Maroon" title="c2">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:53"/><span class="kw">then </span>
<font color="Maroon" title="c2">g</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> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_0.html#T19" target="_self" title="ASYMPT_0:th.19">Th19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:53"/>
<font color="Maroon" title="c2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K12" title="ASYMPT_0:func.12">Big_Theta</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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