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

<b>assume </b><a NAME="E1:25"/><i><font color="Green">E29</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/>

<b>let </b><font color="Maroon">c</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>, <font color="Maroon">seq1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:25"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a>
 <b>and </b><br/><a NAME="E3:25"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">seq1</font> is <a href="seq_2.html#V4">convergent</a>
 ;<br/>

<b>let </b><font color="Maroon">rseq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E4:25"/><i><font color="Green">E36</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">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</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">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</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"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E6:25"/><i><font color="Green">E38</font></i>: 
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">b</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E7:25"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">b</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="txt" href="lp_space.html#E2:25"><i><font color="Green">E31</font></i></a>, , </i>;<br/>
<a NAME="E8:25"/><i><font color="Green">E45</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</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"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="txt" href="lp_space.html#E2:25"><i><font color="Green">E31</font></i></a>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><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>thus </b><font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<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">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<a NAME="E10:25"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">rseq</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">b</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq1</font>
 
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>
<a NAME="E11:25"/><i><font color="Green">E57</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">rseq</font> <a href="hidden.html#R1">=</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"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq1</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="lp_space.html#E3:25"><i><font color="Green">E35</font></i></a>, <a class="txt" href="lp_space.html#E4:25"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="seq_2.html#T20">SEQ_2:20</a></i>;<br/>
<b>thus </b><a NAME="E12:25"/>
<font color="Maroon">rseq</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i>, <a class="txt" href="lp_space.html#E3:25"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="seq_2.html#T19">SEQ_2:19</a></i>;<br/>

<b>thus </b><a NAME="E13:25"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">rseq</font> <a href="hidden.html#R1">=</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"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq1</font></span>)</span>
 <b>by </b><i><a class="txt" href="lp_space.html#E6:25"><i><font color="Green">E38</font></i></a></i>;<br/>


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