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<a NAME="E1:52_1_1"/>
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9" target="_self">Big_Oh</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K10" target="_self">Big_Omega</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:52_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
not <span class="p1">(<span class="default"><a href="asympt_0.html#K9" target="_self">Big_Oh</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K10" target="_self">Big_Omega</a> <font color="Maroon">f</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:52_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K9" target="_self">Big_Oh</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K10" target="_self">Big_Omega</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><a NAME="E2:52_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9" target="_self">Big_Oh</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:52_1_1_1"/><b>then </b>
 ex <font color="Olive">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>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> &amp;  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="Olive">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="Olive">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>hence </b><a NAME="E4:52_1_1_1"/>
<font color="Maroon">x</font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/></div><b>end;</b></div>
<a NAME="E4:52_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="asympt_0.html#K9" target="_self">Big_Oh</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K10" target="_self">Big_Omega</a> <font color="Maroon">f</font></span>)</span> is <a href="fraenkel.html#V1">functional</a>
 
<b>by </b><i><a class="ref" href="fraenkel.html#D1">FRAENKEL:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="asympt_0.html#K9" target="_self">Big_Oh</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K10" target="_self">Big_Omega</a> <font color="Maroon">f</font></span>)</span>;<br/><a NAME="E1:52_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9" target="_self">Big_Oh</a> <font color="Maroon">f</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="E2:52_1_1_2"/>
<font color="Maroon">x</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:52_1_1"/>
<span class="p1">(<span class="default"><a href="asympt_0.html#K9" target="_self">Big_Oh</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K10" target="_self">Big_Omega</a> <font color="Maroon">f</font></span>)</span> is    <a href="fraenkel.html#M1">FUNCTION_DOMAIN</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i>, , <a class="ref" href="fraenkel.html#D2">FRAENKEL:def 2</a></i>;<br/>


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