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<div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">i</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>



<b>thus </b><span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">i</font></span>)</span> <a href="funct_2.html#K3">.</a> 0 = 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <font color="Maroon">i</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">i</font></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="txt" href="real_3.html#E3:52"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="int_1.html#T47">INT_1:47</a></i>
;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( <span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <font color="Maroon">i</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">i</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 );<br/>
<a NAME="E2:52"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_2"/>
<span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <font color="Maroon">i</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="real_3.html#T3" target="_self">Th3</a></i>;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">i</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <font color="Maroon">i</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default">0</span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="real_3.html#T3" target="_self">Th3</a></i>
<br/>.= 
0

;<br/>


</div><b>end;</b></div>
<a NAME="E3:52"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:52_3"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <font color="Maroon">i</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> = 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="int_1.html#K4">frac</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <font color="Maroon">i</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E3:52"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">0 <a href="real_1.html#K5">-</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#T6" target="_self">Th6</a>, <a class="ref" href="int_1.html#T47">INT_1:47</a></i>
<br/>.= 
0

;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">i</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> = 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <font color="Maroon">i</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="txt" href="real_3.html#E1:52_3"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default">0</span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="real_3.html#T3" target="_self">Th3</a></i>
<br/>.= 
0

;<br/>


</div><b>end;</b></div>
<a NAME="E4:52"/>
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>(, <a class="ref" href="real_3.html#T5" target="_self">Th5</a>);<br/></i>
<b>hence </b><a NAME="E5:52"/>
<span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">i</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div>
