let seq1, seq2 be Functional_Sequence of (REAL m),REAL; ( seq1 . 0 = f | Z & ( for i being Nat holds seq1 . (i + 1) = (seq1 . i) `partial| (Z,(I /. (i + 1))) ) & seq2 . 0 = f | Z & ( for i being Nat holds seq2 . (i + 1) = (seq2 . i) `partial| (Z,(I /. (i + 1))) ) implies seq1 = seq2 )
assume that
A3:
seq1 . 0 = f | Z
and
A4:
for n being Nat holds seq1 . (n + 1) = (seq1 . n) `partial| (Z,(I /. (n + 1)))
and
A5:
seq2 . 0 = f | Z
and
A6:
for n being Nat holds seq2 . (n + 1) = (seq2 . n) `partial| (Z,(I /. (n + 1)))
; seq1 = seq2
defpred S1[ Nat] means seq1 . $1 = seq2 . $1;
A7:
for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be
Nat;
( S1[k] implies S1[k + 1] )
assume A8:
S1[
k]
;
S1[k + 1]
seq1 . (k + 1) = (seq1 . k) `partial| (
Z,
(I /. (k + 1)))
by A4;
hence
seq1 . (k + 1) = seq2 . (k + 1)
by A6, A8;
verum
end;
A9:
S1[ 0 ]
by A3, A5;
for n being Nat holds S1[n]
from NAT_1:sch 2(A9, A7);
hence
seq1 = seq2
; verum