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<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>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><span class="p1">(<span class="default"><a href="diff_1.html#NK12" target="_self">bdif</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>
<a NAME="E1:24"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i/>;<br/>
<a NAME="E2:24"/><i><font color="Green">E41</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="24_1"><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="E1:24_1"/>
<span class="p1">(<span class="default"><a href="diff_1.html#NK12" target="_self">bdif</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">k</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> 
 ;<br/>

<a NAME="E2:24_1"/><b>then </b>
 <a href="diff_1.html#K5" target="_self">bD</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK12" target="_self">bdif</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">h</font> 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/>
<b>hence </b><a NAME="E3:24_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:24"/>
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="E4:24"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="diff_1.html#NK12" target="_self">bdif</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> 
 ;<br/>


</div>
