<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">r</font>, <font color="Maroon" title="c3">h</font>, <font color="Maroon" title="c4">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c5">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>









<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>;<br/>
<a NAME="E1:16"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<a NAME="E1:16_1"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
;<br/>
<a NAME="E2:16_1"/>
<font color="Maroon" title="c6">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
;<br/>
<a NAME="E3:16_1"/><b>then </b><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:16_1"/><i><font color="Green" title="E13">A4</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="diff_1.html#E1:16_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> = 
<span class="p1">(<span class="default"><a href="diff_1.html#K3" title="DIFF_1:func.3">fD</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="ref" href="diff_1.html#D6" target="_self" title="DIFF_1:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="diff_1.html#K4" title="DIFF_1:func.4">fD</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="ref" href="diff_1.html#D6" target="_self" title="DIFF_1:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">h</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
<b>by </b><i><a class="ref" href="diff_1.html#T3" target="_self" title="DIFF_1:th.3">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">h</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
<b>by </b><i><a class="txt" href="diff_1.html#E3:16_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">h</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="diff_1.html#E4:16_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">h</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#K4" title="DIFF_1:func.4">fD</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
<b>by </b><i><a class="ref" href="diff_1.html#T3" target="_self" title="DIFF_1:th.3">Th3</a></i>
<br/>.= 
<font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#K3" title="DIFF_1:func.3">fD</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
<b>by </b><i><a class="ref" href="diff_1.html#D6" target="_self" title="DIFF_1:def.6">Def6</a></i>
<br/>.= 
<font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
<b>by </b><i><a class="ref" href="diff_1.html#D6" target="_self" title="DIFF_1:def.6">Def6</a></i>
;<br/>
<b>hence </b><a NAME="E6:16_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:16"/><i><font color="Green" title="E11">A5</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:16_2"/><i><font color="Green" title="E12">A6</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c7">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<a NAME="E2:16_2"/><i><font color="Green" title="E13">A7</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="diff_1.html#E1:16_2"><i><font color="Green" title="E12">A6</font></i></a></i>;<br/>
<a NAME="E3:16_2"/><i><font color="Green" title="E14">A8</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">h</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="diff_1.html#E1:16_2"><i><font color="Green" title="E12">A6</font></i></a></i>;<br/>
<a NAME="E4:16_2"/><i><font color="Green" title="E15">A9</font></i>: 
( <span class="p1">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <span class="p1">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 
<b>by </b><i><a class="ref" href="diff_1.html#T2" target="_self" title="DIFF_1:th.2">Th2</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
<span class="p1">(<span class="default"><a href="diff_1.html#K3" title="DIFF_1:func.3">fD</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<b>by </b><i><a class="ref" href="diff_1.html#D6" target="_self" title="DIFF_1:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">h</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="diff_1.html#E2:16_2"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="diff_1.html#E3:16_2"><i><font color="Green" title="E14">A8</font></i></a>, <a class="txt" href="diff_1.html#E4:16_2"><i><font color="Green" title="E15">A9</font></i></a>, <a class="ref" href="diff_1.html#T3" target="_self" title="DIFF_1:th.3">Th3</a></i>
<br/>.= 
<font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">h</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#K3" title="DIFF_1:func.3">fD</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
<b>by </b><i><a class="txt" href="diff_1.html#E4:16_2"><i><font color="Green" title="E15">A9</font></i></a>, <a class="ref" href="diff_1.html#T3" target="_self" title="DIFF_1:th.3">Th3</a></i>
<br/>.= 
<font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
<b>by </b><i><a class="ref" href="diff_1.html#D6" target="_self" title="DIFF_1:def.6">Def6</a></i>
;<br/>
<b>hence </b><a NAME="E6:16_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:16"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="diff_1.html#E1:16"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="diff_1.html#E2:16"><i><font color="Green" title="E11">A5</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:16"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK10" title="DIFF_1:NK.10">fdif</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
 ;<br/>


</div>
