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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</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>given </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E1:69"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 <b>and </b><br/><a NAME="E2:69"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#R2" target="_self">is_smooth_wrt</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E3:69"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="asympt_0.html#V6" target="_self">eventually-nondecreasing</a> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K15" target="_self">taken_every</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9" target="_self">Big_Oh</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="asympt_0.html#E2:69"><i><font color="Green">E35</font></i></a>, <a class="ref" href="asympt_0.html#D19" target="_self">Def19</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</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="E5:69"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K15" target="_self">taken_every</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E6:69"/><i><font color="Green">E38</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="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</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>4</sub></font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:69"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E9:69"/><i><font color="Green">E40</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>5</sub></font> holds <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</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#E6:69"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:69"/><i><font color="Green">E41</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>6</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#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E3:69"><i><font color="Green">E36</font></i></a>, <a class="ref" href="asympt_0.html#D8" target="_self">Def8</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>
<a NAME="E12:69"/><i><font color="Green">E42</font></i>: 
(  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:69_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/><a NAME="E2:69_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K15" target="_self">taken_every</a> <font color="Maroon">c<sub>8</sub></font> is    <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>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="int_1.html#K2">\]</a></span>;<br/><a NAME="E3:69_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="asympt_0.html#E1:69"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:69_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_0.html#E1:69"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E5:69_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_0.html#E1:69"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E6:69_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="asympt_0.html#E1:69"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E7:69_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="asympt_0.html#E1:69_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:69_1"/><b>then </b>
 <a href="power.html#K6">log</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="power.html#K6">log</a> <font color="Maroon">c<sub>2</sub></font>,1
 <b>by </b><i><a class="txt" href="asympt_0.html#E3:69_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="power.html#T65">POWER:65</a></i>;<br/><a NAME="E9:69_1"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="power.html#K6">log</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_0.html#E4:69_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="asympt_0.html#E6:69_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="power.html#T59">POWER:59</a></i>;<br/><a NAME="E10:69_1"/><i><font color="Green">E50</font></i>: 
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="power.html#K6">log</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/><a NAME="E11:69_1"/><b>then </b>
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="asympt_0.html#E9:69_1"><i><font color="Green">E49</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="int_1.html#K2">\]</a></span> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E13:69_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E8:69"><i><font color="Green">E39</font></i></a>, <a class="ref" href="power.html#T48">POWER:48</a></i>;<br/><a NAME="E14:69_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_0.html#E8:69"><i><font color="Green">E39</font></i></a>, <a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="newton.html#T99">NEWTON:99</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:69_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>12</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>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:69_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="asympt_0.html#E12:69"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:69_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="asympt_0.html#E1:69_1"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E2:69_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E4:69_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="asympt_0.html#E6:69_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="power.html#D3">POWER:def 3</a></i>;<br/>
<a NAME="E3:69_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E3:69_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="asympt_0.html#E10:69_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="pre_ff.html#T10">PRE_FF:10</a></i>;<br/>
<b>hence </b><a NAME="E4:69_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E5:69_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="asympt_0.html#E2:69_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="power.html#T48">POWER:48</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:69_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E3:69_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E5:69_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="asympt_0.html#E1:69_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:69_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E7:69_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E2:69_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:69_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E11:69"><i><font color="Green">E41</font></i></a>, <a class="txt" href="asympt_0.html#E4:69_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="asympt_0.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E9:69_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K15" target="_self">taken_every</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="asympt_0.html#D18" target="_self">Def18</a></i>;<br/><a NAME="E10:69_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K15" target="_self">taken_every</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</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#K15" target="_self">taken_every</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="asympt_0.html#D18" target="_self">Def18</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</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>12</sub></font></span>)</span>;<br/><a NAME="E11:69_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_2"><b>proof </b></a><div class="add">

<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> = 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="power.html#K4">to_power</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_0.html#E5:69_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="power.html#T48">POWER:48</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T29">POWER:29</a></i>
<br/>.= 
1 <a href="real_1.html#K4">*</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>12</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="power.html#K4">to_power</a> 0</span>)</span> <a href="real_1.html#K4">*</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>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T29">POWER:29</a></i>
;<br/>
<b>hence </b><a NAME="E2:69_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="txt" href="asympt_0.html#E8:69"><i><font color="Green">E39</font></i></a>, <a class="ref" href="power.html#T48">POWER:48</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:69_1_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>set </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font>;<br/>
<b>assume </b><a NAME="E1:69_1_1_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</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>12</sub></font></span>)</span>
 ;<br/>

<i><font color="Green">E61</font></i>: <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> = 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_0.html#E5:69_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="power.html#T48">POWER:48</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="power.html#K4">to_power</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_0.html#E4:69_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_0.html#E5:69_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="power.html#T48">POWER:48</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>

;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="newton.html#T99">NEWTON:99</a></i>;<br/>
<a NAME="E4:69_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E6:69_1_1_3"/>
<font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_3_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_1_1_3_2_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_3_2_1_1"><b>suppose </b></a><a NAME="E1:69_1_1_3_2_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="power.html#K4">to_power</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="asympt_0.html#E5:69_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="power.html#T48">POWER:48</a></i>
<br/>.= 
1 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="power.html#T29">POWER:29</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font>

;<br/><b>hence </b><a NAME="E3:69_1_1_3_2_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E12:69"><i><font color="Green">E42</font></i></a>, <a class="txt" href="asympt_0.html#E1:69_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_3_2_1_2"><b>suppose </b></a><a NAME="E1:69_1_1_3_2_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:69_1_1_3_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="asympt_0.html#E3:69_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="power.html#T40">POWER:40</a></i>;<br/><a NAME="E3:69_1_1_3_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="asympt_0.html#E5:69_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="power.html#T48">POWER:48</a></i>;<br/><a NAME="E4:69_1_1_3_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E5:69_1_1_3_2_1_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E2:69_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E7:69_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K15" target="_self">taken_every</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</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>16</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E5:69"><i><font color="Green">E37</font></i></a>, <a class="txt" href="asympt_0.html#E9:69"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E8:69_1_1_3"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</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>16</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="asympt_0.html#D18" target="_self">Def18</a></i>;<br/>
<a NAME="E9:69_1_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</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>16</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</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>4</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</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>12</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E8:69"><i><font color="Green">E39</font></i></a>, <a class="txt" href="asympt_0.html#E1:69_1_1_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E10:69_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</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>16</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</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>12</sub></font></span>)</span>
 
;<br/>
<a NAME="E11:69_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</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>16</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</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>12</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E8:69"><i><font color="Green">E39</font></i></a>, <a class="ref" href="power.html#T48">POWER:48</a></i>;<br/>
<a NAME="E12:69_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</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>16</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="power.html#K4">to_power</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</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>12</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/>
<a NAME="E13:69_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</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>16</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</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>12</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E8:69"><i><font color="Green">E39</font></i></a>, <a class="ref" href="power.html#T32">POWER:32</a></i>;<br/>
<a NAME="E14:69_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</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>16</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</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>12</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E8:69"><i><font color="Green">E39</font></i></a>, <a class="ref" href="power.html#T48">POWER:48</a></i>;<br/>
<b>hence </b><a NAME="E15:69_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="asympt_0.html#E2:69_1_1_3"><i><font color="Green">E61</font></i></a>, <a class="txt" href="asympt_0.html#E8:69_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:69_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="asympt_0.html#E11:69_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="asympt_0.html#E12:69_1_1"><i><font color="Green">E59</font></i></a>);<br/></i><a NAME="E14:69_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</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>12</sub></font></span>)</span>
 ;<br/><a NAME="E15:69_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K15" target="_self">taken_every</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</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>12</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="asympt_0.html#D18" target="_self">Def18</a></i>;<br/><b>hence </b><a NAME="E16:69_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K15" target="_self">taken_every</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</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>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E10:69_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E17:69_1_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</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>12</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="asympt_0.html#E9:69"><i><font color="Green">E40</font></i></a>, <a class="txt" href="asympt_0.html#E2:69_1_1"><i><font color="Green">E53</font></i></a></i>;<br/><div><i><font color="Green">E61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_1_1_4"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><a NAME="E2:69_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</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>12</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> 0
 <b>by </b><i><a class="txt" href="asympt_0.html#E8:69"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E3:69_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="asympt_0.html#E17:69_1_1"><i><font color="Green">E60</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E19:69_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E11:69"><i><font color="Green">E41</font></i></a>, <a class="txt" href="asympt_0.html#E2:69_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="asympt_0.html#E6:69_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="asympt_0.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E20:69_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K15" target="_self">taken_every</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="asympt_0.html#E18:69_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="asympt_0.html#D18" target="_self">Def18</a></i>;<br/></div><b>end;</b></div><a NAME="E17:69_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#K15" target="_self">taken_every</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9" target="_self">Big_Oh</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E2:69_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="asympt_0.html#E14:69_1"><i><font color="Green">E51</font></i></a></i>;<br/><b>hence </b><a NAME="E18:69_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="asympt_0.html#R2" target="_self">is_smooth_wrt</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E3:69"><i><font color="Green">E36</font></i></a>, <a class="ref" href="asympt_0.html#D19" target="_self">Def19</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:69"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="asympt_0.html#V7" target="_self">smooth</a>
 <b>by </b><i><a class="ref" href="asympt_0.html#D20" target="_self">Def20</a></i>;<br/>


</div>
