<?xml version="1.0"?>
<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>let </b><font color="Maroon">h</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>





<b>assume </b><a NAME="E1:34"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V3">constant</a>
 ;<br/>

<a NAME="E2:34"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds  <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:34_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 
;<br/>
<a NAME="E2:34_1"/>
<font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 
;<br/>
<a NAME="E3:34_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:34_1"/>
<font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:34_1"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D16">FUNCT_1:def 16</a>, , </i>;<br/>
<b>hence </b><a NAME="E6:34_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div><b>end;</b></div>
<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="real_1.html#NM1">Real</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK14" target="_self">cdif</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> 0;<br/>
<a NAME="E3:34"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

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

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK14" target="_self">cdif</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="diff_1.html#K7" target="_self">cD</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK14" target="_self">cdif</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> 0</span>)</span>,<font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="diff_1.html#K8" target="_self">cD</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
0
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:34"/><i><font color="Green">E49</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">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3"><b>proof </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="E1:34_3"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK14" target="_self">cdif</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

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

<a NAME="E2:34_3"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK14" target="_self">cdif</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:34_3"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><a href="diff_1.html#NK14" target="_self">cdif</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i/>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK14" target="_self">cdif</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK14" target="_self">cdif</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="diff_1.html#K7" target="_self">cD</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK14" target="_self">cdif</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK14" target="_self">cdif</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK14" target="_self">cdif</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
0
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E5:34_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK14" target="_self">cdif</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:34"/>
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>(, );<br/></i>
<b>hence </b><a NAME="E6:34"/>
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK14" target="_self">cdif</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div>
