<?xml version="1.0"?>
<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">a<sub>1</sub></font></span>)</span><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">a<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">a<sub>1</sub></font></span>)</span><br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="euclid.html#K1">REAL</a> <font color="Maroon">a<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> holds <br/>( ( <font color="Olive">b<sub>3</sub></font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) iff for <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>6</sub></font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>for <font color="Olive">b<sub>7</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <font color="Olive">b<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">b<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>6</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Olive">b<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Olive">b<sub>6</sub></font> ) );<br/>
<a NAME="E1:21"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> 0</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> 0;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> 0</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="euclid.html#K1">REAL</a> 0;<br/>







<b>assume </b><a NAME="E1:21_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 0 holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Olive">b<sub>2</sub></font> )
 ;<br/><b>thus </b><a NAME="E2:21_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>proof </b></a><div class="add">

<font color="Maroon">c<sub>2</sub></font> = 
 <a href="euclid.html#K16">0.REAL</a> 0
<b>by </b><i><a class="ref" href="jordan2c.html#T113">JORDAN2C:113</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
 <a href="euclid.html#K5">0*</a> 0
<b>by </b><i><a class="ref" href="euclid.html#D9">EUCLID:def 9</a></i>
;<br/>
<a NAME="E2:21_1_1_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> 0</span>)</span>
 
<b>by </b><i><a class="ref" href="real_ns1.html#D4" target="_self">Def4</a>, <a class="txt" href="real_ns1.html#E1:21_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E3:21_1_1_1"/><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>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> 0</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:21_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 0
 
<b>by </b><i><a class="ref" href="jordan2c.html#T113">JORDAN2C:113</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E2:21_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> 0
 
<b>by </b><i><a class="ref" href="euclid.html#D9">EUCLID:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:21_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> 0</span>)</span>
 <b>by </b><i><a class="txt" href="real_ns1.html#E1:21_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="real_ns1.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:21_1_1_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:21_1_1_1_3"/><i><font color="Green">E23</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>take </b><font color="Maroon">c<sub>6</sub></font> = 1;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E2:21_1_1_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> = 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p4">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p4">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> 0</span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="txt" href="real_ns1.html#E2:21_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="real_ns1.html#E3:21_1_1_1"><i><font color="Green">E21</font></i></a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p3">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> 0</span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="normsp_1.html#T5">NORMSP_1:5</a></i>
;<br/>
<b>hence </b><a NAME="E4:21_1_1_1_3"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="real_ns1.html#E1:21_1_1_1_3"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:21_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="normsp_1.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="normsp_1.html#D9">NORMSP_1:def 9</a></i>;<br/>

<b>hence </b><a NAME="E6:21_1_1_1"/>
 <a href="normsp_1.html#K8">lim</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="real_ns1.html#E4:21_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="normsp_1.html#D11">NORMSP_1:def 11</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:21_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> ) iff for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 0 holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Olive">b<sub>2</sub></font> ) )
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:21_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</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="euclid.html#K1">REAL</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/><b>assume </b><a NAME="E1:21_2_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><div><i><font color="Green">E22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_2_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1_1"><b>now </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/><b>assume </b><a NAME="E1:21_2_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E3:21_2_1_1_1"/><i><font color="Green">E25</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="seq_1.html#K2">.</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="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i><a NAME="E4:21_2_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>7</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1_1_1"><b>proof </b></a><div class="add">

<div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:21_2_1_1_1_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>thus </b><a NAME="E2:21_2_1_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> holds <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>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:21_2_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> is   <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>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:21_2_1_1_1_1_1_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="real_ns1.html#E1:21_2_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="real_ns1.html#E1:21_2_1_1_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="normsp_1.html#D11">NORMSP_1:def 11</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E4:21_2_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E5:21_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="real_ns1.html#E3:21_2_1_1_1_1_1_1"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E6:21_2_1_1_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="euclid.html#T2">EUCLID:2</a></i>;<br/>
<a NAME="E7:21_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="txt" href="real_ns1.html#E1:21_2_1_1_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E8:21_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T47">RVSUM_1:47</a></i>;<br/>
<a NAME="E9:21_2_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="real_ns1.html#E1:21_2_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="real_ns1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E10:21_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E31</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>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="real_ns1.html#T9" target="_self">Th9</a>, <a class="txt" href="real_ns1.html#E1:21_2_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="real_ns1.html#E8:21_2_1_1_1_1_1_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E11:21_2_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="real_ns1.html#E3:21_2_1_1_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:21_2_1_1_1_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="txt" href="real_ns1.html#E5:21_2_1_1_1_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="real_ns1.html#E10:21_2_1_1_1_1_1_1"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E2:21_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E3:21_2_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="real_ns1.html#E1:21_2_1_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:21_2_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="real_ns1.html#E3:21_2_1_1_1"><i><font color="Green">E25</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:21_2_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Olive">b<sub>2</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_2_1_2"/><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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Olive">b<sub>2</sub></font> )
 ;<br/><b>thus </b><a NAME="E2:21_2_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_2_1"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font>] means <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:21_2_1_2_1"/><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>ex <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> 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>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_2_1_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/>

<b>take </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:21_2_1_2_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="euclid.html#T2">EUCLID:2</a></i>;<br/>
<a NAME="E2:21_2_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E3:21_2_1_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E4:21_2_1_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:21_2_1_2_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>] )
 <b>by </b><i><a class="ref" href="real_ns1.html#T6" target="_self">Th6</a></i>;<br/>


</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="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:21_2_1_2_1"/><i><font color="Green">E25</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>6</sub></font> <a href="real_ns1.html#K5" target="_self">.</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="real_ns1.html#E1:21_2_1_2_1"><i><font color="Green">E24</font></i></a>);<br/></i>
<a NAME="E4:21_2_1_2_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="real_ns1.html#D4" target="_self">Def4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="normsp_1.html#NM2">sequence</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<a NAME="E6:21_2_1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="euclid.html#T2">EUCLID:2</a></i>;<br/>
<a NAME="E7:21_2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E8:21_2_1_2_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E9:21_2_1_2_1"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="real_ns1.html#T6" target="_self">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="real_ns1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E12:21_2_1_2_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 
;<br/>
<a NAME="E13:21_2_1_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:21_2_1_2_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:21_2_1_2_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E3:21_2_1_2_1_2"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<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="E5:21_2_1_2_1_2"/><i><font color="Green">E28</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>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <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="real_ns1.html#E1:21_2_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="real_ns1.html#E1:21_2_1_2_1_2"><i><font color="Green">E27</font></i></a></i>;<br/>
<div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>thus </b><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="real_ns1.html#E5:21_2_1_2_1_2"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="real_ns1.html#E1:21_2_1_2_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="real_ns1.html#E3:21_2_1_2_1"><i><font color="Green">E25</font></i></a></i>
;<br/></div><b>end;</b></div>
<font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="real_ns1.html#E1:21_2_1_2_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<b>hence </b><a NAME="E8:21_2_1_2_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="real_ns1.html#E5:21_2_1_2_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="real_ns1.html#E6:21_2_1_2_1_2"><i><font color="Green">E29</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:21_2_1_2_1"/><b>then </b><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="real_ns1.html#E12:21_2_1_2_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="real_ns1.html#E1:21_2"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E15:21_2_1_2_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<b>then consider </b><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="E17:21_2_1_2_1"/><i><font color="Green">E28</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>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="real_ns1.html#E1:21_2_1_2"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E18:21_2_1_2_1"/><i><font color="Green">E29</font></i>: 
( <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="real_ns1.html#E14:21_2_1_2_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="normsp_1.html#T41">NORMSP_1:41</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span>;<br/>
<a NAME="E19:21_2_1_2_1"/><i><font color="Green">E30</font></i>: 
<span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#T28">SEQ_2:28</a>, <a class="txt" href="real_ns1.html#E18:21_2_1_2_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<i><font color="Green">E31</font></i>:  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> = 
0 <a href="binop_2.html#K24">*</a> 0
<b>by </b><i><a class="ref" href="seq_2.html#T29">SEQ_2:29</a>, <a class="txt" href="real_ns1.html#E18:21_2_1_2_1"><i><font color="Green">E29</font></i></a></i>
<br/>.= 
0

;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  =  <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">a<sub>1</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E22:21_2_1_2_1"/><i><font color="Green">E32</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>12</sub></font> <a href="seq_1.html#K2">.</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="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E23:21_2_1_2_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_2_1_4"><b>proof </b></a><div class="add">

<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_2_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:21_2_1_2_1_4_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:21_2_1_2_1_4_1"/><i><font color="Green">E36</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>14</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"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="real_ns1.html#E1:21_2_1_2_1_4_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a>, <a class="txt" href="real_ns1.html#E17:21_2_1_2_1"><i><font color="Green">E28</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_2_1_4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:21_2_1_2_1_4_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><a NAME="E2:21_2_1_2_1_4_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</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>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="real_ns1.html#E3:21_2_1_2_1_4_1"><i><font color="Green">E36</font></i></a></i>;<br/><b>hence </b><a NAME="E3:21_2_1_2_1_4_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="real_ns1.html#E22:21_2_1_2_1"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:21_2_1_2_1_4_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> holds <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>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/></div><b>end;</b></div>
<a NAME="E2:21_2_1_2_1_4"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E3:21_2_1_2_1_4"/>
( <font color="Maroon">c<sub>12</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="real_ns1.html#E1:21_2_1_2_1_4"><i><font color="Green">E34</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K11">(#)</a> <font color="Maroon">c<sub>12</sub></font>;<br/>
<a NAME="E24:21_2_1_2_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K11">(#)</a> <font color="Maroon">c<sub>12</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#T28">SEQ_2:28</a>, <a class="txt" href="real_ns1.html#E23:21_2_1_2_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<i><font color="Green">E35</font></i>:  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K11">(#)</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> = 
0 <a href="binop_2.html#K24">*</a> 0
<b>by </b><i><a class="ref" href="seq_2.html#T29">SEQ_2:29</a>, <a class="txt" href="real_ns1.html#E23:21_2_1_2_1"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
0

;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span>;<br/>
<a NAME="E26:21_2_1_2_1"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K11">(#)</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_2_1_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_2_1_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><i><font color="Green">E37</font></i>: <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font> = 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#D10">NORMSP_1:def 10</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#D7">NORMSP_1:def 7</a></i>
;<br/><i><font color="Green">E38</font></i>: <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font> = 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#D10">NORMSP_1:def 10</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#D7">NORMSP_1:def 7</a></i>
;<br/><a NAME="E3:21_2_1_2_1_6_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="real_ns1.html#E1:21_2_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="real_ns1.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E4:21_2_1_2_1_6_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 <b>by </b><i><a class="ref" href="real_ns1.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E5:21_2_1_2_1_6_1"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="real_ns1.html#T5" target="_self">Th5</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/><i><font color="Green">E41</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T48">RVSUM_1:48</a></i>
<br/>.= 
<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>15</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="real_ns1.html#E17:21_2_1_2_1"><i><font color="Green">E28</font></i></a></i>
;<br/><a NAME="E9:21_2_1_2_1_6_1"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_2_1_6_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:21_2_1_2_1_6_1_4"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="euclid.html#T2">EUCLID:2</a></i>;<br/>
<a NAME="E2:21_2_1_2_1_6_1_4"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E3:21_2_1_2_1_6_1_4"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E4:21_2_1_2_1_6_1_4"/><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="euclid.html#T2">EUCLID:2</a></i>;<br/>
<a NAME="E5:21_2_1_2_1_6_1_4"/><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="real_ns1.html#E3:21_2_1_2_1_6_1_4"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:21_2_1_2_1_6_1_4"/><i><font color="Green">E46</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="real_ns1.html#E4:21_2_1_2_1_6_1_4"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_2_1_6_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:21_2_1_2_1_6_1_4_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E2:21_2_1_2_1_6_1_4_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="real_ns1.html#E1:21_2_1_2_1_6_1_4_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="real_ns1.html#E3:21_2_1_2_1_6_1_4"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/><a NAME="E5:21_2_1_2_1_6_1_4_1"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T48">RVSUM_1:48</a></i>;<br/><a NAME="E6:21_2_1_2_1_6_1_4_1"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T48">RVSUM_1:48</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="real_ns1.html#E2:21_2_1_2_1_6_1_4_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</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>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="real_ns1.html#E2:21_2_1_2_1_6_1_4_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="real_ns1.html#E5:21_2_1_2_1_6_1_4_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</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>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="real_ns1.html#E2:21_2_1_2_1_6_1_4_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</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>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</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>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="real_ns1.html#E6:21_2_1_2_1_6_1_4_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="real_ns1.html#E3:21_2_1_2_1"><i><font color="Green">E25</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:21_2_1_2_1_6_1_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="real_ns1.html#E5:21_2_1_2_1_6_1_4"><i><font color="Green">E45</font></i></a>, <a class="txt" href="real_ns1.html#E6:21_2_1_2_1_6_1_4"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:21_2_1_2_1_6_1"/><i><font color="Green">E43</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="square_1.html#T72">SQUARE_1:72</a></i>;<br/><a NAME="E11:21_2_1_2_1_6_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font> <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>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="real_ns1.html#E22:21_2_1_2_1"><i><font color="Green">E32</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i><a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a>, <a class="txt" href="real_ns1.html#E4:21_2_1_2_1_6_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="real_ns1.html#E1:21_2_1_2_1_6_1"><i><font color="Green">E37</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="binop_2.html#K9">+</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>15</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="real_ns1.html#T10" target="_self">Th10</a>, <a class="txt" href="real_ns1.html#E8:21_2_1_2_1_6_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="real_ns1.html#E9:21_2_1_2_1_6_1"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="binop_2.html#K9">+</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>15</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="real_ns1.html#E5:21_2_1_2_1_6_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="real_ns1.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="binop_2.html#K11">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="binop_2.html#K9">+</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>15</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="binop_2.html#K9">+</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>15</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a>, <a class="txt" href="real_ns1.html#E2:21_2_1_2_1_6_1"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="binop_2.html#K9">+</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>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="real_ns1.html#E10:21_2_1_2_1_6_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="binop_2.html#K9">+</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>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="binop_2.html#K9">+</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>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <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>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K11">(#)</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></span>)</span>
<b>by </b><i><a class="txt" href="real_ns1.html#E11:21_2_1_2_1_6_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:21_2_1_2_1_6"/>
<span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K11">(#)</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:21_2_1_2_1"/><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#T19">SEQ_2:19</a>, <a class="txt" href="real_ns1.html#E19:21_2_1_2_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="real_ns1.html#E24:21_2_1_2_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="real_ns1.html#E26:21_2_1_2_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<i><font color="Green">E38</font></i>:  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> = 
0 <a href="binop_2.html#K23">+</a> 0
<b>by </b><i><a class="ref" href="seq_2.html#T20">SEQ_2:20</a>, <a class="txt" href="real_ns1.html#E19:21_2_1_2_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="real_ns1.html#E20:21_2_1_2_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="real_ns1.html#E24:21_2_1_2_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="real_ns1.html#E25:21_2_1_2_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="real_ns1.html#E26:21_2_1_2_1"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
0

;<br/>
<a NAME="E29:21_2_1_2_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_2_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:21_2_1_2_1_8"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:21_2_1_2_1_8"/>
<font color="Maroon">c<sub>15</sub></font> <a href="binop_2.html#K11">*</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>15</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="real_ns1.html#E1:21_2_1_2_1_8"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xreal_1.html#T99">XREAL_1:99</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:21_2_1_2_1_8"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>16</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"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p4"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>15</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="real_ns1.html#E27:21_2_1_2_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="real_ns1.html#E28:21_2_1_2_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_2_1_8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:21_2_1_2_1_8_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/><i><font color="Green">E43</font></i>: <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> = 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#D10">NORMSP_1:def 10</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#D7">NORMSP_1:def 7</a></i>
;<br/><a NAME="E3:21_2_1_2_1_8_1"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="txt" href="real_ns1.html#E2:21_2_1_2_1_8_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>;<br/><a NAME="E4:21_2_1_2_1_8_1"/><i><font color="Green">E45</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="normsp_1.html#T8">NORMSP_1:8</a></i>;<br/><a NAME="E5:21_2_1_2_1_8_1"/><b>then </b><i><font color="Green">E46</font></i>: 
0 <a href="binop_2.html#K11">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E6:21_2_1_2_1_8_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"><span class="p4"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p4"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="txt" href="real_ns1.html#E5:21_2_1_2_1_8_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="real_ns1.html#E3:21_2_1_2_1_8_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E7:21_2_1_2_1_8_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>15</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="real_ns1.html#E4:21_2_1_2_1_8"><i><font color="Green">E41</font></i></a>, <a class="txt" href="real_ns1.html#E1:21_2_1_2_1_8_1"><i><font color="Green">E42</font></i></a></i>;<br/><i><font color="Green">E48</font></i>:  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> = 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="real_ns1.html#E1:21_2_1_2_1_8"><i><font color="Green">E40</font></i></a>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>
;<br/> <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="binop_2.html#K11">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> = 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="txt" href="real_ns1.html#E4:21_2_1_2_1_8_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>
;<br/><b>hence </b><a NAME="E10:21_2_1_2_1_8_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="square_1.html#T95">SQUARE_1:95</a>, <a class="txt" href="real_ns1.html#E5:21_2_1_2_1_8_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="real_ns1.html#E7:21_2_1_2_1_8_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="real_ns1.html#E8:21_2_1_2_1_8_1"><i><font color="Green">E48</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:21_2_1_2_1_8"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E30:21_2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is <a href="normsp_1.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="normsp_1.html#D9">NORMSP_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E31:21_2_1_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="real_ns1.html#E29:21_2_1_2_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="normsp_1.html#D11">NORMSP_1:def 11</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:21_2_1"/>
( ( <font color="Maroon">c<sub>4</sub></font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> ) iff for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Olive">b<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="real_ns1.html#E2:21_2_1"><i><font color="Green">E22</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:21_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1]
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E3:21"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="real_ns1.html#E1:21"><i><font color="Green">E18</font></i></a>, <a class="txt" href="real_ns1.html#E2:21"><i><font color="Green">E19</font></i></a>);</i><br/>


</div>
