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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">x</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">xs</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">seq</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">xseq</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">xs</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Olive">xseq</font> <a href="hidden.html#R1">=</a> <font color="Olive">seq</font> holds <br/>( ( <font color="Olive">seq</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Olive">seq</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> ) iff for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</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">rseqi</font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">rseqi</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">rseqi</font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Olive">xs</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Olive">rseqi</font> ) );<br/>
<a NAME="E1:21"/><i><font color="Green">E33</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">x</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">xs</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">seq</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">xseq</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">E34</font></i>: 
( <font color="Maroon">xs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">xseq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq</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">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 0 holds <br/> ex <font color="Olive">rseqi</font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">rseqi</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">rseqi</font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Olive">rseqi</font> )
 ;<br/><b>thus </b><a NAME="E2:21_1_1"/>
( <font color="Maroon">seq</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</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">xs</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">E53</font></i>: 
<font color="Maroon">x</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>, </i>;<br/>
<a NAME="E3:21_1_1_1"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">i</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">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:21_1_1_1_2"/>
<font color="Maroon">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">i</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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">i</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</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>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:21_1_1_1"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> holds <br/> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</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">r</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">E67</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/>

<b>take </b><font color="Maroon">m</font> = 1;<br/>

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:21_1_1_1_3"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</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>, </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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:21_1_1_1"/>
<font color="Maroon">seq</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">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <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">seq</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) iff for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 0 holds <br/> ex <font color="Olive">rseqi</font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">rseqi</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">rseqi</font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Olive">rseqi</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">E70</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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:21_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</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">x</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">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">xs</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">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/><b>let </b><font color="Maroon">seq</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">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">xseq</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">n</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">E72</font></i>: 
( <font color="Maroon">xs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">xseq</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">seq</font> )
 ;<br/><div><i><font color="Green">E73</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">E74</font></i>: 
( <font color="Maroon">seq</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</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">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:21_2_1_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">i</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">n</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="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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">xseq</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">i</font>;<br/><b>consider </b><font color="Maroon">rseqi</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">E78</font></i>: 
for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">rseqi</font> <a href="seq_1.html#K2">.</a> <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</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">rseqi</font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">rseqi</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">E80</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">e</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">E88</font></i>: 
<font color="Maroon">e</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">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</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">rseqi</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</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">e</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">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:21_2_1_1_1_1_1_1"/><i><font color="Green">E89</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, , <a class="ref" href="normsp_1.html#D11">NORMSP_1:def 11</a></i>;<br/>
<b>take </b>
<font color="Maroon">k</font>
;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

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

<a NAME="E5:21_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E90</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="txt" href="real_ns1.html#E1"><i><font color="Green">Lemma28</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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</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">i</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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xs</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, </i>;<br/>
<a NAME="E8:21_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xs</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xs</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:21_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E105</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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E11:21_2_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">rseqi</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><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">rseqi</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</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">rseqi</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">rseqi</font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">rseqi</font> )
 <b>by </b><i>, <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">rseqi</font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">rseqi</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">rseqi</font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Olive">rseqi</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:21_2_1_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</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">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> holds <br/> ex <font color="Olive">rseqi</font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">rseqi</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">rseqi</font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Olive">rseqi</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">E106</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</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">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> holds <br/> ex <font color="Olive">rseqi</font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">rseqi</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">rseqi</font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Olive">rseqi</font> )
 ;<br/><b>thus </b><a NAME="E2:21_2_1_2"/>
( <font color="Maroon">seq</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_2_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xseq</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">n</font>;<br/>
<a NAME="E1:21_2_1_2_1"/><i><font color="Green">E108</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">y</font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>,<font color="Olive">y</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">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>take </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><font color="Maroon">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</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">n</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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">i</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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</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">y</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">y</font>] )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">xseqn</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">n</font><b> such that </b><br/><a NAME="E3:21_2_1_2_1"/><i><font color="Green">E110</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>,<font color="Maroon">xseqn</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</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">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">seqn</font> = <font color="Maroon">xseqn</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">n</font></span>)</span> ;<br/>
<a NAME="E6:21_2_1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">xs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</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">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">xs</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">xs</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</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">xs</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</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">xsn</font> = <font color="Maroon">xs</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">xn</font> = <font color="Maroon">xsn</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">n</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E12:21_2_1_2_1"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">xsn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xn</font> &amp; <font color="Maroon">xseqn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seqn</font> )
 
;<br/>
<a NAME="E13:21_2_1_2_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/> ex <font color="Olive">rseqi</font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">rseqi</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xseqn</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">rseqi</font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">xsn</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Olive">rseqi</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">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:21_2_1_2_1_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:21_2_1_2_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</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">n</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">n</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">rseqi</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">E115</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Maroon">rseqi</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">rseqi</font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">rseqi</font> )
 <b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E116</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">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><font color="Maroon">rseqi</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="real_ns1.html#E1"><i><font color="Green">Lemma28</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</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">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</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">xseqn</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<font color="Maroon">xsn</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">xs</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">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <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">rseqi</font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">rseqi</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xseqn</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">rseqi</font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">xsn</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Olive">rseqi</font> )
 <b>by </b><i><a class="txt" href="real_ns1.html#E1"><i><font color="Green">Lemma28</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">E117</font></i>: 
( <font color="Maroon">seqn</font> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Maroon">xn</font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">seqn</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E15:21_2_1_2_1"/>
<font color="Maroon">n</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">n</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">rseqn1</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">E119</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Maroon">rseqn1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp; <font color="Maroon">rseqn1</font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</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">rseqn1</font> )
 <b>by </b><i/>;<br/>
<a NAME="E18:21_2_1_2_1"/><i><font color="Green">E120</font></i>: 
( <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</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">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</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="ref" href="normsp_1.html#T41">NORMSP_1:41</a></i>;<br/>
<b>set </b><font color="Maroon">seqn2</font> = <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</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">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</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">E122</font></i>: 
<span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</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">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</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>, </i>;<br/>
<i><font color="Green">E123</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">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</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">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</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>, </i>
<br/>.= 
0

;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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">rseqn1</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">xs</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">absrseq</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">E125</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">absrseq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</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">E126</font></i>: 
( <font color="Maroon">absrseq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">absrseq</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">E127</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">e</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">E128</font></i>: 
<font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:21_2_1_2_1_4_1"/><i><font color="Green">E129</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</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">rseqn1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</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">e</font>
 <b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a>, <a class="txt" href="real_ns1.html#E1"><i><font color="Green">Lemma28</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">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:21_2_1_2_1_4_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</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">rseqn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</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">e</font>
 <b>by </b><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">absrseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K10">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:21_2_1_2_1_4_1"/>
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</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">absrseq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K10">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E2:21_2_1_2_1_4"/><b>then </b>
<font color="Maroon">absrseq</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">absrseq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">absrseq</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, <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">rseqn2</font> = <font color="Maroon">absrseq</font> <a href="seq_1.html#K11">(#)</a> <font color="Maroon">absrseq</font>;<br/>
<a NAME="E24:21_2_1_2_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">absrseq</font> <a href="seq_1.html#K11">(#)</a> <font color="Maroon">absrseq</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#E1:21"><i><font color="Green">E33</font></i></a></i>;<br/>
<i><font color="Green">E132</font></i>:  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">absrseq</font> <a href="seq_1.html#K11">(#)</a> <font color="Maroon">absrseq</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#E1:21"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
0

;<br/>
<b>set </b><font color="Maroon">seq2</font> = <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</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">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</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">E134</font></i>: 
<span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</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">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</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">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</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">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</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">absrseq</font> <a href="seq_1.html#K11">(#)</a> <font color="Maroon">absrseq</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">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><i><font color="Green">E135</font></i>: <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</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">E136</font></i>: <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</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">seqn</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xn</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xs</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:21_2_1_2_1_6_1"/><b>then </b><i><font color="Green">E137</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xs</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 <b>by </b><i/>;<br/><a NAME="E5:21_2_1_2_1_6_1"/><i><font color="Green">E138</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">seqn</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xn</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xseqn</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xsn</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">rxseqk</font> = <font color="Maroon">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">n</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">rxs</font> = <font color="Maroon">xs</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">n</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">E141</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xs</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">rxseqk</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</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">rxs</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</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">rseqn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="real_ns1.html#E1"><i><font color="Green">Lemma28</font></i></a></i>
;<br/><a NAME="E9:21_2_1_2_1_6_1"/><i><font color="Green">E142</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xs</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xseqn</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xsn</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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</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">n</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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xs</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">E143</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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xs</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</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">E144</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">xseqn</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xsn</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</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">E145</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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xs</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <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">E146</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">xseqn</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xsn</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="real_ns1.html#D1" target="_self">Def1</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">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:21_2_1_2_1_6_1_4_1"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">i</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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xs</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E2:21_2_1_2_1_6_1_4_1"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">xseq1</font> = <font color="Maroon">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font>, <font color="Maroon">xs1</font> = <font color="Maroon">xs</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">n</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">xseq2</font> = <font color="Maroon">xseqn</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font>, <font color="Maroon">xs2</font> = <font color="Maroon">xsn</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</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">E153</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xs</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xseq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">xs1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</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">E154</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xseqn</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xsn</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xseq2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">xs2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xs</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xs</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">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xs</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="real_ns1.html#D2" target="_self">Def2</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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</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">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="real_ns1.html#D2" target="_self">Def2</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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</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">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</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">xs</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">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="real_ns1.html#D2" target="_self">Def2</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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</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">xs</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">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</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">xs</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</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">xseqn</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xsn</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i>, </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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xs</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xseqn</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xsn</font>
 <b>by </b><i>, , <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">E155</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">rseqn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</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">E156</font></i>: 
<font color="Maroon">absrseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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">rseqn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><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">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</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">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xs</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#E2"><i><font color="Green">Lemma39</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">xseqn</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">xsn</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">rseqn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</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>, , </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">seqn</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xn</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">rseqn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</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>, </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">seqn</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xn</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">seqn</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xn</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">rseqn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</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">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</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">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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">rseqn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</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>, </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">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</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">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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">rseqn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</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="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">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</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">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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">rseqn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</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">rseqn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</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">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</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">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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">rseqn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</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">rseqn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</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">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</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">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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">absrseq</font> <a href="seq_1.html#K11">(#)</a> <font color="Maroon">absrseq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="real_ns1.html#D1" target="_self">Def1</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">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</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">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</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">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</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">seqn</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">xn</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">absrseq</font> <a href="seq_1.html#K11">(#)</a> <font color="Maroon">absrseq</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">E157</font></i>: 
<span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</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">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</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>, , , </i>;<br/>
<i><font color="Green">E158</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">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</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">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</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>, , , , , </i>
<br/>.= 
0

;<br/>
<a NAME="E29:21_2_1_2_1"/><i><font color="Green">E159</font></i>: 
for <font color="Olive">e</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">m</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">e</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">e</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">E160</font></i>: 
<font color="Maroon">e</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">e</font> <a href="binop_2.html#K11">*</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">e</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T99">XREAL_1:99</a></i>;<br/>
<b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:21_2_1_2_1_8"/><i><font color="Green">E161</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</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">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</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">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K10">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">e</font>
 <b>by </b><i>, , <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">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:21_2_1_2_1_8_1"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><i><font color="Green">E163</font></i>: <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</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">E164</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">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</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">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, <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">E165</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</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">E166</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</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">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</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">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, <a class="ref" href="real_ns1.html#D1" target="_self">Def1</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">E167</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">e</font>
 <b>by </b><i>, </i>;<br/><i><font color="Green">E168</font></i>:  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">e</font></span>)</span> = 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon">e</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">e</font>
<b>by </b><i>, <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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i>, <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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="square_1.html#T95">SQUARE_1:95</a>, , , <a class="ref" href="real_ns1.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:21_2_1_2_1_8"/>
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">m</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E30:21_2_1_2_1"/><b>then </b>
<font color="Maroon">seq</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">seq</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="real_ns1.html#E2"><i><font color="Green">Lemma39</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">seq</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) iff for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</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">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> holds <br/> ex <font color="Olive">rseqi</font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">rseqi</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">rseqi</font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">xs</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Olive">rseqi</font> ) )
 <b>by </b><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">n</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">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


</div>
