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

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">h</font>, <font color="Olive" title="b2">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b3">f1</font>, <font color="Olive" title="b4">f2</font> being  <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>  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="Olive" title="b3">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b4">f2</font></span>)</span>,<font color="Olive" title="b1">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="Olive" title="b2">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="Olive" title="b3">f1</font>,<font color="Olive" title="b1">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="Olive" title="b2">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="Olive" title="b4">f2</font>,<font color="Olive" title="b1">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="Olive" title="b2">x</font></span>)</span></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being  <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>  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="Olive" title="b1">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">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#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="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="Olive" title="b1">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#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="c3">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="Olive" title="b2">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#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="c3">x</font></span>)</span></span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="valued_1.html#K3" title="VALUED_1:func.3">+</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#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="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#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="c3">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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#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="c3">x</font></span>)</span></span><br/>









<span class="kw">defpred </span><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> ] <span class="kw">means </span>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="valued_1.html#K3" title="VALUED_1:func.3">+</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#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> <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#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> <a href="real_1.html#K7" title="REAL_1:func.7">+</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#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="E1:27"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
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="27_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:27_1"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
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="valued_1.html#K3" title="VALUED_1:func.3">+</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#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> <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#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> <a href="real_1.html#K7" title="REAL_1:func.7">+</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#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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="valued_1.html#K3" title="VALUED_1:func.3">+</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#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> <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#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> <a href="real_1.html#K7" title="REAL_1:func.7">+</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#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></span><br/>

<a NAME="E2:27_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
( <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="valued_1.html#K3" title="VALUED_1:func.3">+</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#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> <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#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> <a href="real_1.html#K7" title="REAL_1:func.7">+</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#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> &amp; <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="valued_1.html#K3" title="VALUED_1:func.3">+</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#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#K9" title="REAL_1:func.9">-</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#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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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#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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>;<br/>
<a NAME="E3:27_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<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="valued_1.html#K3" title="VALUED_1:func.3">+</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#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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diff_1.html#T12" target="_self" title="DIFF_1:th.12">Th12</a></span>;<br/>
<a NAME="E4:27_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<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#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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diff_1.html#T12" target="_self" title="DIFF_1:th.12">Th12</a></span>;<br/>
<a NAME="E5:27_1"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<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#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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diff_1.html#T12" target="_self" title="DIFF_1:th.12">Th12</a></span>;<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="valued_1.html#K3" title="VALUED_1:func.3">+</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#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#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="valued_1.html#K3" title="VALUED_1:func.3">+</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#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="diff_1.html#D7" target="_self" title="DIFF_1:def.7">Def7</a></span>
<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="valued_1.html#K3" title="VALUED_1:func.3">+</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#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> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="valued_1.html#K3" title="VALUED_1:func.3">+</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#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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="diff_1.html#T4" target="_self" title="DIFF_1:th.4">Th4</a></span>
<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#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> <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#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#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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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#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> <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#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#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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a></span>
<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#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">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> <a href="real_1.html#K7" title="REAL_1:func.7">+</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#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> <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#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#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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="diff_1.html#T4" target="_self" title="DIFF_1:th.4">Th4</a></span>
<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#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">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> <a href="real_1.html#K7" title="REAL_1:func.7">+</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#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="diff_1.html#T4" target="_self" title="DIFF_1:th.4">Th4</a></span>
<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#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> <a href="real_1.html#K7" title="REAL_1:func.7">+</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#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="diff_1.html#D7" target="_self" title="DIFF_1:def.7">Def7</a></span>
<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#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> <a href="real_1.html#K7" title="REAL_1:func.7">+</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#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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="diff_1.html#D7" target="_self" title="DIFF_1:def.7">Def7</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:27_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="valued_1.html#K3" title="VALUED_1:func.3">+</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#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> <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#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> <a href="real_1.html#K7" title="REAL_1:func.7">+</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#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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:27"/><span class="lab"><font color="Green" title="E17">A7</font></span>: 
<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="27_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="valued_1.html#K3" title="VALUED_1:func.3">+</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"><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> <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"><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> <a href="real_1.html#K7" title="REAL_1:func.7">+</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"><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></span><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="valued_1.html#K3" title="VALUED_1:func.3">+</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"><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#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="valued_1.html#K3" title="VALUED_1:func.3">+</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c2">h</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="diff_1.html#D7" target="_self" title="DIFF_1:def.7">Def7</a></span>
<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="valued_1.html#K3" title="VALUED_1:func.3">+</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="diff_1.html#D7" target="_self" title="DIFF_1:def.7">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="diff_1.html#T4" target="_self" title="DIFF_1:th.4">Th4</a></span>
<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#K7" title="REAL_1:func.7">+</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#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></span>
<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#K7" title="REAL_1:func.7">+</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#K9" title="REAL_1:func.9">-</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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></span>
<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#K9" title="REAL_1:func.9">-</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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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#K9" title="REAL_1:func.9">-</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#K9" title="REAL_1:func.9">-</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#K7" title="REAL_1:func.7">+</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#K9" title="REAL_1:func.9">-</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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="diff_1.html#T4" target="_self" title="DIFF_1:th.4">Th4</a></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#K7" title="REAL_1:func.7">+</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="diff_1.html#T4" target="_self" title="DIFF_1:th.4">Th4</a></span>
<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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c2">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> <a href="real_1.html#K7" title="REAL_1:func.7">+</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="diff_1.html#D7" target="_self" title="DIFF_1:def.7">Def7</a></span>
<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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c2">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> <a href="real_1.html#K7" title="REAL_1:func.7">+</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c2">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="diff_1.html#D7" target="_self" title="DIFF_1:def.7">Def7</a></span>
<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"><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> <a href="real_1.html#K7" title="REAL_1:func.7">+</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c2">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="diff_1.html#D7" target="_self" title="DIFF_1:def.7">Def7</a></span>
<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"><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> <a href="real_1.html#K7" title="REAL_1:func.7">+</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"><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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="diff_1.html#D7" target="_self" title="DIFF_1:def.7">Def7</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:27_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="valued_1.html#K3" title="VALUED_1:func.3">+</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"><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> <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"><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> <a href="real_1.html#K7" title="REAL_1:func.7">+</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"><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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:27"/>
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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E16">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:27"/>
<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="valued_1.html#K3" title="VALUED_1:func.3">+</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#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="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#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="c3">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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#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="c3">x</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
