let seq1, seq2 be Functional_Sequence of REAL ,REAL ; :: thesis: ( seq1 . 0 = f & ( for n being Element of NAT holds seq1 . (n + 1) = fD (seq1 . n),h ) & seq2 . 0 = f & ( for n being Element of NAT holds seq2 . (n + 1) = fD (seq2 . n),h ) implies seq1 = seq2 )
assume that
A7: ( seq1 . 0 = f & ( for n being Element of NAT holds seq1 . (n + 1) = fD (seq1 . n),h ) ) and
A8: ( seq2 . 0 = f & ( for n being Element of NAT holds seq2 . (n + 1) = fD (seq2 . n),h ) ) ; :: thesis: seq1 = seq2
defpred S₁[ Nat] means seq1 . $1 = seq2 . $1;
A9: S₁[ 0 ] by A7, A8;
A10: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A11: S₁[k] ; :: thesis: S₁[k + 1]
thus seq1 . (k + 1) = fD (seq1 . k),h by A7
.= seq2 . (k + 1) by A8, A11 ; :: thesis: verum

end;

for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A9, A10);
hence seq1 = seq2 by FUNCT_2:113; :: thesis: verum