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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="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  <a href="rcomp_1.html#V1">compact</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:18"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:18"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>and </b><br/><a NAME="E3:18"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E4:18"/><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> holds <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="jct_misc.html#V1" target="_self">connected</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,  <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ] means ( <font color="Maroon">a<sub>2</sub></font> is <a href="integra1.html#V1">closed-interval</a> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E5:18"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:18_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E4:18"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:18_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E4:18_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="jct_misc.html#E4:18_1"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E6:18_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E4:18"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:18_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E9:18_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="jct_misc.html#E9:18_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E11:18_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> is <a href="jct_misc.html#V1" target="_self">connected</a>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E4:18"><i><font color="Green">E18</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1"><b>suppose </b></a><a NAME="E1:18_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><div class="add"><b>take </b>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
;<br/><b>thus </b><a NAME="E2:18_1_1_1"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> is <a href="integra1.html#V1">closed-interval</a>
 <b>by </b><i><a class="txt" href="jct_misc.html#E1:18_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="integra1.html#D1">INTEGRA1:def 1</a></i>;<br/><a NAME="E3:18_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E1:18_1_1_1"><i><font color="Green">E25</font></i></a></i>;<br/><b>hence </b><a NAME="E4:18_1_1_1"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E4:18_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E5:18_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E1:18_1_1_1"><i><font color="Green">E25</font></i></a></i>;<br/><b>hence </b><a NAME="E6:18_1_1_1"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E9:18_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>thus </b><a NAME="E7:18_1_1_1"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E3:18_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jct_misc.html#E8:18_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jct_misc.html#E11:18_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="jct_misc.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2"><b>suppose </b></a><a NAME="E1:18_1_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><b>take </b>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
;<br/><b>thus </b><a NAME="E2:18_1_1_2"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> is <a href="integra1.html#V1">closed-interval</a>
 <b>by </b><i><a class="txt" href="jct_misc.html#E1:18_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="integra1.html#D1">INTEGRA1:def 1</a></i>;<br/><a NAME="E3:18_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E1:18_1_1_2"><i><font color="Green">E25</font></i></a></i>;<br/><b>hence </b><a NAME="E4:18_1_1_2"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E4:18_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E5:18_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E1:18_1_1_2"><i><font color="Green">E25</font></i></a></i>;<br/><b>hence </b><a NAME="E6:18_1_1_2"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E9:18_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>thus </b><a NAME="E7:18_1_1_2"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E3:18_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jct_misc.html#E8:18_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jct_misc.html#E11:18_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="jct_misc.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E7:18"/><i><font color="Green">E20</font></i>: 
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>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="jct_misc.html#E5:18"><i><font color="Green">E19</font></i></a>);<br/></i>
<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="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E9:18"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<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="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E11:18"/><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> holds <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,  <a href="real_1.html#NM1">Real</a>,  <a href="real_1.html#NM1">Real</a>] means ( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> &amp;  <a href="jct_misc.html#K1" target="_self">dist</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">a<sub>3</sub></font></span>)</span> );<br/>
<a NAME="E12:18"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> as  <a href="integra1.html#V1">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="jct_misc.html#E7:18"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E2:18_2"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="jct_misc.html#E7:18"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E3:18_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="jct_misc.html#E9:18"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jct_misc.html#E11:18"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="rcomp_1.html#V1">compact</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="jct_misc.html#E2:18_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="jct_misc.html#T14" target="_self">Th14</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E6:18_2"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="jct_misc.html#K1" target="_self">dist</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="jct_misc.html#T18" target="_self">Th18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="real_1.html#NM1">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/>

<b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/>

<b>thus </b><a NAME="E8:18_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>]
 <b>by </b><i><a class="txt" href="jct_misc.html#E6:18_2"><i><font color="Green">E25</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E14:18"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="jct_misc.html#S2" target="_self">JCT_MISC:sch 2</a>(<a class="txt" href="jct_misc.html#E12:18"><i><font color="Green">E23</font></i></a>);<br/></i>
<a NAME="E15:18"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:18_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:18_3"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E4:18_3"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="jct_misc.html#E3:18_3"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E6:18_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E14:18"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E7:18_3"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E9:18"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jct_misc.html#E4:18_3"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E8:18_3"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E9:18_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E2:18"><i><font color="Green">E16</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E17:18"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E18:18"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="seq_2.html#V4">convergent</a>
 <b>and </b><br/><a NAME="E19:18"/><i><font color="Green">E27</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D3">RCOMP_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E21:18"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E17:18"><i><font color="Green">E25</font></i></a>, <a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>;<br/>
<a NAME="E22:18"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:18_4"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:18_4"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:18_4"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="jct_misc.html#E3:18_4"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E6:18_4"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>;<br/>
<a NAME="E7:18_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <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>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E14:18"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E8:18_4"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E11:18"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jct_misc.html#E4:18_4"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E9:18_4"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E10:18_4"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E3:18"><i><font color="Green">E17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E24:18"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>14</sub></font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">c<sub>10</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E25:18"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>14</sub></font> is <a href="seq_2.html#V4">convergent</a>
 <b>and </b><br/><a NAME="E26:18"/><i><font color="Green">E31</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D3">RCOMP_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E28:18"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E24:18"><i><font color="Green">E29</font></i></a>, <a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2;<br/>
<a NAME="E29:18"/>
<font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/>
<a NAME="E30:18"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E18:18"><i><font color="Green">E26</font></i></a>, <a class="ref" href="seq_4.html#T30">SEQ_4:30</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>18</sub></font> =  <a href="jct_misc.html#K1" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>;<br/>
<a NAME="E31:18"/>
<font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/>
<a NAME="E32:18"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E18:18"><i><font color="Green">E26</font></i></a>, <a class="ref" href="seq_4.html#T29">SEQ_4:29</a></i>;<br/>
<a NAME="E33:18"/><b>then </b><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>14</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E25:18"><i><font color="Green">E30</font></i></a>, <a class="ref" href="seq_2.html#T19">SEQ_2:19</a></i>;<br/>
<a NAME="E34:18"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#T21">SEQ_2:21</a></i>;<br/>
<i><font color="Green">E37</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jct_misc.html#E25:18"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jct_misc.html#E32:18"><i><font color="Green">E34</font></i></a>, <a class="ref" href="seq_2.html#T20">SEQ_2:20</a></i>
<br/>.= 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jct_misc.html#E33:18"><i><font color="Green">E35</font></i></a>, <a class="ref" href="seq_2.html#T22">SEQ_2:22</a></i>
;<br/>
<a NAME="E36:18"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="jct_misc.html#E7:18"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E37:18"/><b>then </b>
( not <font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">c<sub>4</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/>
<a NAME="E38:18"/><b>then </b>
 <a href="jct_misc.html#K1" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="jct_misc.html#E1:18"><i><font color="Green">E15</font></i></a>, <a class="ref" href="jct_misc.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E39:18"/><b>then </b>
<span class="p1">(<span class="default"><a href="jct_misc.html#K1" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E41:18"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="jct_misc.html#K1" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i><a class="txt" href="jct_misc.html#E34:18"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jct_misc.html#E35:18"><i><font color="Green">E37</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_6"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 2

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
;<br/><a NAME="E3:18_6"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="jct_misc.html#K1" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i><a class="txt" href="jct_misc.html#E41:18"><i><font color="Green">E38</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>22</sub></font> =  <a href="jct_misc.html#K1" target="_self">dist</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E4:18_6"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="jct_misc.html#E7:18"><i><font color="Green">E20</font></i></a></i>;<br/><a NAME="E5:18_6"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E6:18_6"/><b>then </b><i><font color="Green">E42</font></i>: 
( not <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a> )
 <b>by </b><i><a class="txt" href="jct_misc.html#E9:18"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jct_misc.html#E11:18"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E7:18_6"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="jct_misc.html#E9:18"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jct_misc.html#E11:18"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E8:18_6"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E9:18_6"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="jct_misc.html#K1" target="_self">dist</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="jct_misc.html#K1" target="_self">dist</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E6:18_6"><i><font color="Green">E42</font></i></a>, <a class="ref" href="jct_misc.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E10:18_6"/><b>then </b>
<span class="p1">(<span class="default"><a href="jct_misc.html#K1" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jct_misc.html#K1" target="_self">dist</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E11:18_6"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" target="_self">dist</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" target="_self">dist</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" target="_self">dist</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E12:18_6"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" target="_self">dist</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jct_misc.html#K1" target="_self">dist</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/><i><font color="Green">E46</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> = 
<font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jct_misc.html#E21:18"><i><font color="Green">E28</font></i></a>, <a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>
;<br/><i><font color="Green">E47</font></i>: <font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jct_misc.html#E28:18"><i><font color="Green">E32</font></i></a>, <a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>
;<br/><a NAME="E15:18_6"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> is <a href="integra1.html#V1">closed-interval</a>
 <b>by </b><i><a class="txt" href="jct_misc.html#E7:18"><i><font color="Green">E20</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>23</sub></font>, <font color="Maroon">c<sub>24</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E17:18_6"/>
<font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font>
 <b>and </b><br/><a NAME="E18:18_6"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="integra1.html#D1">INTEGRA1:def 1</a></i>;<br/><a NAME="E19:18_6"/>
2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="jct_misc.html#K1" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E20:18_6"/><b>then </b><i><font color="Green">E49</font></i>: 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jct_misc.html#K1" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E3:18_6"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jct_misc.html#E13:18_6"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jct_misc.html#E14:18_6"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><i><font color="Green">E50</font></i>: 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>

;<br/><a NAME="E22:18_6"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jct_misc.html#E14:18"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E23:18_6"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jct_misc.html#E7:18_6"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E24:18_6"/><i><font color="Green">E53</font></i>: 
 <a href="jct_misc.html#K1" target="_self">dist</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E22:18_6"><i><font color="Green">E51</font></i></a>, <a class="ref" href="jct_misc.html#T16" target="_self">Th16</a></i>;<br/><div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_6_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_6_5_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> or <font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_6_5_1_1"><b>suppose </b></a><a NAME="E1:18_6_5_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:18_6_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E3:18_6_5_1_1"/><b>then </b>
 <a href="jct_misc.html#K1" target="_self">dist</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E24:18_6"><i><font color="Green">E53</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E4:18_6_5_1_1"/><b>then </b>
 <a href="jct_misc.html#K1" target="_self">dist</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E9:18_6"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:18_6_5_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E20:18_6"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jct_misc.html#E21:18_6"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:18_6_5_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T9">RCOMP_1:9</a></i>;<br/><a NAME="E7:18_6_5_1_1"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E18:18_6"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jct_misc.html#E22:18_6"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jct_misc.html#E23:18_6"><i><font color="Green">E52</font></i></a>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/><b>hence </b><a NAME="E8:18_6_5_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E6:18_6_5_1_1"><i><font color="Green">E55</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_6_5_1_2"><b>suppose </b></a><a NAME="E1:18_6_5_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:18_6_5_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E3:18_6_5_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>;<br/><a NAME="E4:18_6_5_1_2"/><b>then </b>
 <a href="jct_misc.html#K1" target="_self">dist</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E24:18_6"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jct_misc.html#E2:18_6_5_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E5:18_6_5_1_2"/><b>then </b>
 <a href="jct_misc.html#K1" target="_self">dist</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E9:18_6"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:18_6_5_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E20:18_6"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jct_misc.html#E21:18_6"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:18_6_5_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T9">RCOMP_1:9</a></i>;<br/><a NAME="E8:18_6_5_1_2"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E18:18_6"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jct_misc.html#E22:18_6"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jct_misc.html#E23:18_6"><i><font color="Green">E52</font></i></a>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/><b>hence </b><a NAME="E9:18_6_5_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E7:18_6_5_1_2"><i><font color="Green">E56</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_6_6"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_6_6"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="txt" href="jct_misc.html#E1:18_6"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_6_6_1"><b>suppose </b></a><a NAME="E1:18_6_6_1"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>

;<br/><a NAME="E3:18_6_6_1"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T142">COMPLEX1:142</a></i>;<br/><i><font color="Green">E57</font></i>:  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T153">COMPLEX1:153</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><a NAME="E5:18_6_6_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E3:18_6"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:18_6_6_1"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E3:18_6_6_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jct_misc.html#E4:18_6_6_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="E7:18_6_6_1"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E14:18"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jct_misc.html#E13:18_6"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E8:18_6_6_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E11:18"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E9:18_6_6_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E25:18_6"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jct_misc.html#E1:18_6_6_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E10:18_6_6_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" target="_self">dist</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E14:18"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jct_misc.html#E13:18_6"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jct_misc.html#E14:18_6"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jct_misc.html#E6:18_6_6_1"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E11:18_6_6_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jct_misc.html#K1" target="_self">dist</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E12:18_6"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E12:18_6_6_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jct_misc.html#E7:18_6_6_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jct_misc.html#E9:18_6_6_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="jct_misc.html#T16" target="_self">Th16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_6_6_2"><b>suppose </b></a><a NAME="E1:18_6_6_2"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>

;<br/><a NAME="E3:18_6_6_2"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T142">COMPLEX1:142</a></i>;<br/><i><font color="Green">E57</font></i>:  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T153">COMPLEX1:153</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><a NAME="E5:18_6_6_2"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E3:18_6"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:18_6_6_2"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E3:18_6_6_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jct_misc.html#E4:18_6_6_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:18_6_6_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E14:18"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jct_misc.html#E14:18_6"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E8:18_6_6_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E9:18"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E9:18_6_6_2"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E25:18_6"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jct_misc.html#E1:18_6_6_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>
<br/>.= 
 <a href="jct_misc.html#K1" target="_self">dist</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jct_misc.html#E14:18"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jct_misc.html#E13:18_6"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jct_misc.html#E14:18_6"><i><font color="Green">E47</font></i></a></i>
;<br/><a NAME="E11:18_6_6_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jct_misc.html#K1" target="_self">dist</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E12:18_6"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jct_misc.html#E6:18_6_6_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E12:18_6_6_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jct_misc.html#E7:18_6_6_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jct_misc.html#E9:18_6_6_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="jct_misc.html#T16" target="_self">Th16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>take </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
;<br/>

<b>thus </b><a NAME="E43:18"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E19:18"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jct_misc.html#E26:18"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jct_misc.html#E30:18"><i><font color="Green">E33</font></i></a>, <a class="ref" href="jct_misc.html#T9" target="_self">Th9</a></i>;<br/>

<b>thus </b><a NAME="E44:18"/>
not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E42:18"><i><font color="Green">E39</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>21</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E45:18"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T33">SEQM_3:33</a></i>;<br/>
<a NAME="E46:18"/>
<font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T33">SEQM_3:33</a></i>;<br/>
<a NAME="E47:18"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E10"><i><font color="Green">Lemma3</font></i></a></i>;<br/>
<a NAME="E48:18"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E45:18"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>22</sub></font> =  <a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font>;<br/>
<a NAME="E49:18"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span>;<br/>

<a NAME="E50:18"/>
<font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E51:18"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E10"><i><font color="Green">Lemma3</font></i></a></i>;<br/>
<a NAME="E52:18"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E10"><i><font color="Green">Lemma3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E53:18"/>
<font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E48:18"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
;<br/>
<a NAME="E55:18"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="jct_misc.html#K1" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i><a class="txt" href="jct_misc.html#E41:18"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jct_misc.html#E49:18"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>24</sub></font> =  <a href="jct_misc.html#K1" target="_self">dist</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>;<br/>
<a NAME="E56:18"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="jct_misc.html#E7:18"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E57:18"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E58:18"/><b>then </b><i><font color="Green">E44</font></i>: 
( not <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 
<b>by </b><i><a class="txt" href="jct_misc.html#E9:18"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jct_misc.html#E11:18"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E59:18"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="jct_misc.html#E9:18"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jct_misc.html#E11:18"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E60:18"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E61:18"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="jct_misc.html#K1" target="_self">dist</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="jct_misc.html#K1" target="_self">dist</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E58:18"><i><font color="Green">E44</font></i></a>, <a class="ref" href="jct_misc.html#T17" target="_self">Th17</a></i>;<br/>
<i><font color="Green">E47</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span>)</span> = 
<font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="txt" href="jct_misc.html#E21:18"><i><font color="Green">E28</font></i></a>, <a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>
;<br/>
<i><font color="Green">E48</font></i>: <font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jct_misc.html#E28:18"><i><font color="Green">E32</font></i></a>, <a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>
;<br/>
<a NAME="E64:18"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> is <a href="integra1.html#V1">closed-interval</a>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E7:18"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>25</sub></font>, <font color="Maroon">c<sub>26</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E66:18"/>
<font color="Maroon">c<sub>25</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font>
 <b>and </b><br/><a NAME="E67:18"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="integra1.html#D1">INTEGRA1:def 1</a></i>;<br/>
<a NAME="E68:18"/>
2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="jct_misc.html#K1" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<a NAME="E69:18"/><b>then </b><i><font color="Green">E50</font></i>: 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jct_misc.html#K1" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E55:18"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jct_misc.html#E62:18"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jct_misc.html#E63:18"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<i><font color="Green">E51</font></i>: 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>

;<br/>
<a NAME="E71:18"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> )
 
<b>by </b><i><a class="txt" href="jct_misc.html#E14:18"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E72:18"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> )
 
<b>by </b><i><a class="txt" href="jct_misc.html#E59:18"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E73:18"/><i><font color="Green">E54</font></i>: 
 <a href="jct_misc.html#K1" target="_self">dist</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E71:18"><i><font color="Green">E52</font></i></a>, <a class="ref" href="jct_misc.html#T16" target="_self">Th16</a></i>;<br/>
<div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_11"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_11_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_11_1_1"><b>suppose </b></a><a NAME="E1:18_11_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/><div class="add"><a NAME="E2:18_11_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E3:18_11_1_1"/><b>then </b>
 <a href="jct_misc.html#K1" target="_self">dist</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E73:18"><i><font color="Green">E54</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E4:18_11_1_1"/><b>then </b>
 <a href="jct_misc.html#K1" target="_self">dist</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E61:18"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:18_11_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E69:18"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jct_misc.html#E70:18"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:18_11_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T9">RCOMP_1:9</a></i>;<br/><a NAME="E7:18_11_1_1"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E67:18"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jct_misc.html#E71:18"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jct_misc.html#E72:18"><i><font color="Green">E53</font></i></a>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/><b>hence </b><a NAME="E8:18_11_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E6:18_11_1_1"><i><font color="Green">E56</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_11_1_2"><b>suppose </b></a><a NAME="E1:18_11_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/><div class="add"><a NAME="E2:18_11_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E3:18_11_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>;<br/><a NAME="E4:18_11_1_2"/><b>then </b>
 <a href="jct_misc.html#K1" target="_self">dist</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E73:18"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jct_misc.html#E2:18_11_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E5:18_11_1_2"/><b>then </b>
 <a href="jct_misc.html#K1" target="_self">dist</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E61:18"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:18_11_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E69:18"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jct_misc.html#E70:18"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:18_11_1_2"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T9">RCOMP_1:9</a></i>;<br/><a NAME="E8:18_11_1_2"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E67:18"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jct_misc.html#E71:18"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jct_misc.html#E72:18"><i><font color="Green">E53</font></i></a>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/><b>hence </b><a NAME="E9:18_11_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E7:18_11_1_2"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E75:18"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E7:18"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>hence </b><a NAME="E76:18"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E74:18"><i><font color="Green">E55</font></i></a></i>;<br/>


</div>
