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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="asympt_0.html#V4" target="_self">eventually-positive</a> <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><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>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:30"/><i><font color="Green">E17</font></i>: 
<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>and </b><br/><a NAME="E2:30"/><i><font color="Green">E18</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#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</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="E4:30"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E5:30"/><i><font color="Green">E20</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>4</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>4</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 ) ) )
 <b>by </b><i><a class="txt" href="asympt_0.html#E1:30"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:30"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E8:30"/><i><font color="Green">E22</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>2</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>5</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>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="txt" href="asympt_0.html#E4:30"><i><font color="Green">E19</font></i></a>, <a class="txt" href="asympt_0.html#E5:30"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> ) </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> ) </span>}</span> ;<br/>
<a NAME="E9:30"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>6</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>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E10:30"/><i><font color="Green">E24</font></i>: 
<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>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E11:30"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  is  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 
<b>from </b><i><a class="ref" href="asympt_0.html#S1" target="_self">ASYMPT_0:sch 1</a>(<a class="txt" href="asympt_0.html#E10:30"><i><font color="Green">E24</font></i></a>);<br/></i>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E13:30"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  is  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 
<b>from </b><i><a class="ref" href="asympt_0.html#S1" target="_self">ASYMPT_0:sch 1</a>(<a class="txt" href="asympt_0.html#E10:30"><i><font color="Green">E24</font></i></a>);<br/></i>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>9</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>10</sub></font>;<br/>
<a NAME="E15:30"/>
 <a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D4">PRE_CIRC:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E17:30"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>13</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>13</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>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<a NAME="E18:30"/><i><font color="Green">E26</font></i>: 
 <a href="seq_4.html#K3">min</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#E2:30"><i><font color="Green">E18</font></i></a>, <a class="txt" href="asympt_0.html#E17:30"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E19:30"/><i><font color="Green">E27</font></i>: 
 <a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="asympt_0.html#E2:30"><i><font color="Green">E18</font></i></a>, <a class="txt" href="asympt_0.html#E17:30"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> =  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>15</sub></font>
;<br/>

<a NAME="E21:30"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>15</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="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<b>thus </b><a NAME="E22:30"/>
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_0.html#E7:30"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E23:30"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E24:30"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="asympt_0.html#E2:30"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E25:30"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="asympt_0.html#E2:30"><i><font color="Green">E18</font></i></a>, <a class="txt" href="asympt_0.html#E23:30"><i><font color="Green">E29</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1"><b>suppose </b></a><a NAME="E1:30_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>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:30_1_1"/><b>then </b>
<font color="Maroon">c<sub>16</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#E23:30"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:30_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <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>5</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#E8:30"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E4:30_1_1"/>
<font color="Maroon">c<sub>5</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>15</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#E21:30"><i><font color="Green">E28</font></i></a>, <a class="txt" href="asympt_0.html#E24:30"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E5:30_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <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>15</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#E3:30_1_1"><i><font color="Green">E32</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="30_1_2"><b>suppose </b></a><a NAME="E1:30_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:30_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <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>15</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_2_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:30_1_2_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:30_1_2_1_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>16</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>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E9:30"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:30_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E23:30"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E4:30_1_2_1_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><a NAME="E5:30_1_2_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="asympt_0.html#E8:30"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E6:30_1_2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E9:30"><i><font color="Green">E23</font></i></a>, <a class="txt" href="asympt_0.html#E1:30_1_2"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E7:30_1_2_1_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><a NAME="E2:30_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E18:30"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E3:30_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="asympt_0.html#E19:30"><i><font color="Green">E27</font></i></a>, <a class="txt" href="asympt_0.html#E5:30_1_2_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="asympt_0.html#E7:30_1_2_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/></div><b>end;</b></div><a NAME="E9:30_1_2_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E23:30"><i><font color="Green">E29</font></i></a>, <a class="txt" href="asympt_0.html#E2:30_1_2_1_1_1"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E10:30_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="pre_circ.html#D4">PRE_CIRC:def 4</a></i>;<br/><a NAME="E11:30_1_2_1_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.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>16</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E8:30_1_2_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_2_1_1_1_2"/>
<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>16</sub></font></span>)</span> <a href="real_1.html#K6">/</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#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 ;<br/><a NAME="E2:30_1_2_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><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>16</sub></font></span>)</span> <a href="real_1.html#K6">/</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>16</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>16</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.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>16</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E24:30"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:30_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.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>16</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E25:30"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E4:30_1_2_1_1_1_2"/><b>then </b>
 <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.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>16</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E4:30_1_2_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:30_1_2_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="asympt_0.html#E19:30"><i><font color="Green">E27</font></i></a>, <a class="txt" href="asympt_0.html#E11:30_1_2_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/></div><b>end;</b></div><a NAME="E13:30_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><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>16</sub></font></span>)</span> <a href="real_1.html#K6">/</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>16</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>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"><a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.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>16</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E24:30"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E14:30_1_2_1_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.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>16</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E25:30"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E15:30_1_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.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>16</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</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#E21:30"><i><font color="Green">E28</font></i></a>, <a class="txt" href="asympt_0.html#E24:30"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E16:30_1_2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <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>15</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#E14:30_1_2_1_1_1"><i><font color="Green">E39</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="30_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:30_1_2_1_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>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:30_1_2_1_1_2"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <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>5</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#E8:30"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E3:30_1_2_1_1_2"/>
<font color="Maroon">c<sub>5</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>15</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#E21:30"><i><font color="Green">E28</font></i></a>, <a class="txt" href="asympt_0.html#E24:30"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E4:30_1_2_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <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>15</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#E2:30_1_2_1_1_2"><i><font color="Green">E33</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></div><b>end;</b></div></div><b>end;</b></div>

</div>
