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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="asympt_0.html#V2" target="_self">eventually-nonnegative</a> <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:51"/><i><font color="Green">E29</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>3</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#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="asympt_0.html#D4" target="_self">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:51"/><i><font color="Green">E30</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>4</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="asympt_0.html#D4" target="_self">Def4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:51_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K10" target="_self">Big_Omega</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K5" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></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> <b> such that </b><br/><a NAME="E3:51_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E4:51_1_1"/><i><font color="Green">E32</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K5" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) ) )
 ;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:51_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E7:51_1_1"/><i><font color="Green">E34</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>8</sub></font> holds <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K5" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="asympt_0.html#E4:51_1_1"><i><font color="Green">E32</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/><a NAME="E8:51_1_1"/><i><font color="Green">E35</font></i>: 
(  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E9:51_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
(  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:51_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/><a NAME="E2:51_1_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="asympt_0.html#E8:51_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="asympt_0.html#E9:51_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:51_1_1_1"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="asympt_0.html#K8" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="asympt_0.html#K2" target="_self">max</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="asympt_0.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E4:51_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="asympt_0.html#E2:51"><i><font color="Green">E29</font></i></a>, <a class="txt" href="asympt_0.html#E2:51_1_1_1"><i><font color="Green">E37</font></i></a></i>;<br/><a NAME="E5:51_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="asympt_0.html#E4:51"><i><font color="Green">E30</font></i></a>, <a class="txt" href="asympt_0.html#E2:51_1_1_1"><i><font color="Green">E37</font></i></a></i>;<br/><a NAME="E6:51_1_1_1"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="asympt_0.html#K8" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K5" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_1_1_1_1_1"/>
( <span class="p1">(<span class="default"><a href="asympt_0.html#K8" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> or <span class="p1">(<span class="default"><a href="asympt_0.html#K8" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> )
 </span><b>by </b><i><a class="txt" href="asympt_0.html#E3:51_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:51_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="asympt_0.html#K8" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><div class="add"><a NAME="E2:51_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_0.html#K8" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E5:51_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E3:51_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="asympt_0.html#K8" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K5" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:51_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="asympt_0.html#K8" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><div class="add"><a NAME="E2:51_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_0.html#K8" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E4:51_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E3:51_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="asympt_0.html#K8" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K5" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E7:51_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K5" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E7:51_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="asympt_0.html#E2:51_1_1_1"><i><font color="Green">E37</font></i></a></i>;<br/><a NAME="E8:51_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_0.html#K8" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K5" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E6:51_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="asympt_0.html#E6:51_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E9:51_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_0.html#K8" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E7:51_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>thus </b><a NAME="E10:51_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="asympt_0.html#E7:51_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="asympt_0.html#E2:51_1_1_1"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E11:51_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K10" target="_self">Big_Omega</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K8" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E3:51_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="asympt_0.html#E6:51_1_1"><i><font color="Green">E33</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:51_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K10" target="_self">Big_Omega</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K8" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></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> <b> such that </b><br/><a NAME="E4:51_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E5:51_1"/><i><font color="Green">E32</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_0.html#K8" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) ) )
 ;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:51_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E8:51_1"/><i><font color="Green">E34</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>8</sub></font> holds <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_0.html#K8" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="asympt_0.html#E5:51_1"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E9:51_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> 0
 <b>by </b><i><a class="txt" href="asympt_0.html#E7:51_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/><a NAME="E10:51_1"/>
(  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E11:51_1"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:51_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/><a NAME="E2:51_1_2"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="asympt_0.html#E11:51_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:51_1_2"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="asympt_0.html#K2" target="_self">max</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="asympt_0.html#K2" target="_self">max</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E4:51_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K2" target="_self">max</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K2" target="_self">max</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E5:51_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K2" target="_self">max</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E6:51_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K2" target="_self">max</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E7:51_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K5" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="asympt_0.html#K2" target="_self">max</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/><a NAME="E8:51_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K5" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K2" target="_self">max</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E7:51_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E9:51_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K5" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_0.html#K8" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="asympt_0.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E10:51_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_0.html#K8" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E8:51_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="asympt_0.html#E2:51_1_2"><i><font color="Green">E37</font></i></a></i>;<br/><b>hence </b><a NAME="E11:51_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K5" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E9:51_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>thus </b><a NAME="E12:51_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="asympt_0.html#E8:51_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="asympt_0.html#E2:51_1_2"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:51_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K10" target="_self">Big_Omega</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K5" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E4:51_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="asympt_0.html#E9:51_1"><i><font color="Green">E35</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:51"/>
 <a href="asympt_0.html#K10" target="_self">Big_Omega</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K5" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="asympt_0.html#K10" target="_self">Big_Omega</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K8" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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