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

<b>let </b><font color="Maroon" title="c3">seq1</font>, <font color="Maroon" title="c4">seq2</font> be    <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">seq1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">f</font> &amp; ( 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" title="c3">seq1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="diff_1.html#K5" title="DIFF_1:func.5">bD</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">h</font> ) &amp; <font color="Maroon" title="c4">seq2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">f</font> &amp; ( 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" title="c4">seq2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="diff_1.html#K5" title="DIFF_1:func.5">bD</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">h</font> ) implies <font color="Maroon" title="c3">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">seq2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:21_1_2"/><i><font color="Green" title="E11">A7</font></i>: 
( <font color="Maroon" title="c3">seq1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">f</font> &amp; ( 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" title="c3">seq1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="diff_1.html#K5" title="DIFF_1:func.5">bD</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">h</font> ) )
 <b>and </b><br/><a NAME="E2:21_1_2"/><i><font color="Green" title="E12">A8</font></i>: 
( <font color="Maroon" title="c4">seq2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">f</font> &amp; ( 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" title="c4">seq2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="diff_1.html#K5" title="DIFF_1:func.5">bD</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">h</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">seq2</font></span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><font color="Maroon" title="c3">seq1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> $1 <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">seq2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> $1;<br/>
<a NAME="E3:21_1_2"/><i><font color="Green" title="E13">A9</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="diff_1.html#E1:21_1_2"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="diff_1.html#E2:21_1_2"><i><font color="Green" title="E12">A8</font></i></a></i>;<br/>
<a NAME="E4:21_1_2"/><i><font color="Green" title="E14">A10</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:21_1_2_1"/><i><font color="Green" title="E15">A11</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<b>thus </b><font color="Maroon" title="c3">seq1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
 <a href="diff_1.html#K5" title="DIFF_1:func.5">bD</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span>,<font color="Maroon" title="c2">h</font>
<b>by </b><i><a class="txt" href="diff_1.html#E1:21_1_2"><i><font color="Green" title="E11">A7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">seq2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="diff_1.html#E2:21_1_2"><i><font color="Green" title="E12">A8</font></i></a>, <a class="txt" href="diff_1.html#E1:21_1_2_1"><i><font color="Green" title="E15">A11</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:21_1_2"/>
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#E3:21_1_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="diff_1.html#E4:21_1_2"><i><font color="Green" title="E14">A10</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:21_1_2"/>
<font color="Maroon" title="c3">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">seq2</font>
 <b>by </b><i><a class="ref" href="seqfunc.html#T2" title="SEQFUNC:th.2">SEQFUNC:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
