<?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">h</font>, <font color="Maroon" title="c3">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">f2</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#NK12" title="DIFF_1:NK.12">bdif</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="partfun3.html#K4" title="PARTFUN3:func.4">-</a> <font color="Maroon" title="c5">f2</font></span>)</span>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font></span>)</span>;<br/>
<a NAME="E1:28"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_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="E1:28_1"/>
<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="E2:28_1"/><b>then </b>
( <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> <font color="Maroon" title="c4">f1</font> &amp; <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> <font color="Maroon" title="c5">f2</font> )
 
<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="E3:28_1"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:28_1"/><b>then </b><i><font color="Green" title="E18">A2</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="c4">f1</font> <a href="partfun3.html#K4" title="PARTFUN3:func.4">-</a> <font color="Maroon" title="c5">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>;<br/>
<a NAME="E5:28_1"/>
<font color="Maroon" title="c6">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">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="E6:28_1"/><b>then </b>
( <font color="Maroon" title="c6">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">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> <font color="Maroon" title="c4">f1</font> &amp; <font color="Maroon" title="c6">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">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> <font color="Maroon" title="c5">f2</font> )
 
<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="E7:28_1"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:28_1"/><b>then </b><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">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="c4">f1</font> <a href="partfun3.html#K4" title="PARTFUN3:func.4">-</a> <font color="Maroon" title="c5">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="partfun3.html#K4" title="PARTFUN3:func.4">-</a> <font color="Maroon" title="c5">f2</font></span>)</span>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font> = 
<span class="p1">(<span class="default"><a href="diff_1.html#K5" title="DIFF_1:func.5">bD</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="partfun3.html#K4" title="PARTFUN3:func.4">-</a> <font color="Maroon" title="c5">f2</font></span>)</span>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> 0</span>)</span>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="ref" href="diff_1.html#D7" target="_self" title="DIFF_1:def.7">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="diff_1.html#K6" title="DIFF_1:func.6">bD</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="partfun3.html#K4" title="PARTFUN3:func.4">-</a> <font color="Maroon" title="c5">f2</font></span>)</span>,<font color="Maroon" title="c2">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#D7" target="_self" title="DIFF_1:def.7">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="partfun3.html#K4" title="PARTFUN3:func.4">-</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="partfun3.html#K4" title="PARTFUN3:func.4">-</a> <font color="Maroon" title="c5">f2</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#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="diff_1.html#T4" target="_self" title="DIFF_1:th.4">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="partfun3.html#K4" title="PARTFUN3:func.4">-</a> <font color="Maroon" title="c5">f2</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#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="diff_1.html#E4:28_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</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#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</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#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="diff_1.html#E8:28_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</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#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</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#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#K6" title="DIFF_1:func.6">bD</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c2">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> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</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#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="diff_1.html#T4" target="_self" title="DIFF_1:th.4">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#K6" title="DIFF_1:func.6">bD</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c2">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> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#K6" title="DIFF_1:func.6">bD</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c2">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#T4" target="_self" title="DIFF_1:th.4">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#K5" title="DIFF_1:func.5">bD</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> 0</span>)</span>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#K6" title="DIFF_1:func.6">bD</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c2">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#D7" target="_self" title="DIFF_1:def.7">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#K5" title="DIFF_1:func.5">bD</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> 0</span>)</span>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#K5" title="DIFF_1:func.5">bD</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> 0</span>)</span>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
<b>by </b><i><a class="ref" href="diff_1.html#D7" target="_self" title="DIFF_1:def.7">Def7</a></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#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#K5" title="DIFF_1:func.5">bD</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> 0</span>)</span>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
<b>by </b><i><a class="ref" href="diff_1.html#D7" target="_self" title="DIFF_1:def.7">Def7</a></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#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
<b>by </b><i><a class="ref" href="diff_1.html#D7" target="_self" title="DIFF_1:def.7">Def7</a></i>
;<br/>
<b>hence </b><a NAME="E10:28_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="partfun3.html#K4" title="PARTFUN3:func.4">-</a> <font color="Maroon" title="c5">f2</font></span>)</span>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:28"/><i><font color="Green" title="E18">A4</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#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_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="E1:28_2"/><i><font color="Green" title="E19">A5</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#NK12" title="DIFF_1:NK.12">bdif</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="partfun3.html#K4" title="PARTFUN3:func.4">-</a> <font color="Maroon" title="c5">f2</font></span>)</span>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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="E2:28_2"/><i><font color="Green" title="E20">A6</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="partfun3.html#K4" title="PARTFUN3:func.4">-</a> <font color="Maroon" title="c5">f2</font></span>)</span>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="diff_1.html#E1:28_2"><i><font color="Green" title="E19">A5</font></i></a></i>;<br/>
<a NAME="E3:28_2"/><i><font color="Green" title="E21">A7</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="partfun3.html#K4" title="PARTFUN3:func.4">-</a> <font color="Maroon" title="c5">f2</font></span>)</span>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="diff_1.html#E1:28_2"><i><font color="Green" title="E19">A5</font></i></a></i>;<br/>
<a NAME="E4:28_2"/><i><font color="Green" title="E22">A8</font></i>: 
<span class="p1">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="partfun3.html#K4" title="PARTFUN3:func.4">-</a> <font color="Maroon" title="c5">f2</font></span>)</span>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</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#T12" target="_self" title="DIFF_1:th.12">Th12</a></i>;<br/>
<a NAME="E5:28_2"/><i><font color="Green" title="E23">A9</font></i>: 
<span class="p1">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</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#T12" target="_self" title="DIFF_1:th.12">Th12</a></i>;<br/>
<a NAME="E6:28_2"/><i><font color="Green" title="E24">A10</font></i>: 
<span class="p1">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</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#T12" target="_self" title="DIFF_1:th.12">Th12</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="partfun3.html#K4" title="PARTFUN3:func.4">-</a> <font color="Maroon" title="c5">f2</font></span>)</span>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font> = 
<span class="p1">(<span class="default"><a href="diff_1.html#K5" title="DIFF_1:func.5">bD</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="partfun3.html#K4" title="PARTFUN3:func.4">-</a> <font color="Maroon" title="c5">f2</font></span>)</span>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font>
<b>by </b><i><a class="ref" href="diff_1.html#D7" target="_self" title="DIFF_1:def.7">Def7</a></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#NK12" title="DIFF_1:NK.12">bdif</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="partfun3.html#K4" title="PARTFUN3:func.4">-</a> <font color="Maroon" title="c5">f2</font></span>)</span>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="partfun3.html#K4" title="PARTFUN3:func.4">-</a> <font color="Maroon" title="c5">f2</font></span>)</span>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="diff_1.html#T4" target="_self" title="DIFF_1:th.4">Th4</a>, <a class="txt" href="diff_1.html#E4:28_2"><i><font color="Green" title="E22">A8</font></i></a></i>
<br/>.= 
<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#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="diff_1.html#E2:28_2"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="diff_1.html#E3:28_2"><i><font color="Green" title="E21">A7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#K5" title="DIFF_1:func.5">bD</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="diff_1.html#T4" target="_self" title="DIFF_1:th.4">Th4</a>, <a class="txt" href="diff_1.html#E5:28_2"><i><font color="Green" title="E23">A9</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#K5" title="DIFF_1:func.5">bD</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#K5" title="DIFF_1:func.5">bD</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
<b>by </b><i><a class="ref" href="diff_1.html#T4" target="_self" title="DIFF_1:th.4">Th4</a>, <a class="txt" href="diff_1.html#E6:28_2"><i><font color="Green" title="E24">A10</font></i></a></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#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#K5" title="DIFF_1:func.5">bD</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
<b>by </b><i><a class="ref" href="diff_1.html#D7" target="_self" title="DIFF_1:def.7">Def7</a></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#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
<b>by </b><i><a class="ref" href="diff_1.html#D7" target="_self" title="DIFF_1:def.7">Def7</a></i>
;<br/>
<b>hence </b><a NAME="E8:28_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="partfun3.html#K4" title="PARTFUN3:func.4">-</a> <font color="Maroon" title="c5">f2</font></span>)</span>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:28"/>
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#E1:28"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="diff_1.html#E2:28"><i><font color="Green" title="E18">A4</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:28"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="partfun3.html#K4" title="PARTFUN3:func.4">-</a> <font color="Maroon" title="c5">f2</font></span>)</span>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c2">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font></span>)</span>
 ;<br/>


</div>
