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<b>let </b><font color="Maroon">s1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">s2</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:105_1_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> 0 &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> ) )
 <b>and </b><br/><a NAME="E2:105_1_2"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">s2</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> 0 &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E3:105_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="ref" href="real_3.html#T14" target="_self">Th14</a>, </i>;<br/>
<a NAME="E4:105_1_2"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:105_1_2_1"/>
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 ;<br/>

<b>hence </b><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#T14" target="_self">Th14</a></i>
<br/>.= 
<font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i/>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E5:105_1_2"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_2.html#D9">FUNCT_2:def 9</a><br/>

<b>thus </b><a NAME="E6:105_1_2"/>
<font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div>
