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<b>let </b><font color="Maroon">f</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:10_3_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">f</font> is <a href="asympt_0.html#V2" target="_self">eventually-nonnegative</a> &amp; <font color="Maroon">f</font> is <a href="asympt_0.html#V5" target="_self">eventually-nonzero</a> )
 ;<br/>

<b>then consider </b><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="E3:10_3_1"/><i><font color="Green">E42</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">f</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/>
<b>consider </b><font color="Maroon">M</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="E5:10_3_1"/><i><font color="Green">E43</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">M</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, </i>;<br/>
<b>take </b><font color="Maroon">a</font> =  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N</font>,<font color="Maroon">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="asympt_0.html#D6" target="_self">ASYMPT_0:def 6</a><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="E6:10_3_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E7:10_3_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">M</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E8:10_3_1"/><b>then </b><i><font color="Green">E56</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">M</font> )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:10_3_1"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:10_3_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/>


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