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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:10_2_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="asympt_0.html#V4" target="_self">eventually-positive</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:10_2_1"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="asympt_0.html#D6" target="_self">Def6</a></i>;<br/>
<b>thus </b><a NAME="E4:10_2_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="asympt_0.html#V2" target="_self">eventually-nonnegative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="asympt_0.html#D4" target="_self">ASYMPT_0:def 4</a><br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:10_2_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:10_2_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="asympt_0.html#E3:10_2_1"><i><font color="Green">E7</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:10_2_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="asympt_0.html#V5" target="_self">eventually-nonzero</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="asympt_0.html#D7" target="_self">ASYMPT_0:def 7</a><br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:10_2_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:10_2_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_0.html#E3:10_2_1"><i><font color="Green">E7</font></i></a></i>;<br/>


</div><b>end;</b></div>


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