let seq1, seq2 be Complex_Sequence; :: thesis: ( seq1 . 0 = 1 & ( for n being Element of NAT holds seq1 . (n + 1) = (seq1 . n) * (n + 1) ) & seq2 . 0 = 1 & ( for n being Element of NAT holds seq2 . (n + 1) = (seq2 . n) * (n + 1) ) implies seq1 = seq2 )
assume that
A2: seq1 . 0 = 1 and
A3: for n being Element of NAT holds seq1 . (n + 1) = (seq1 . n) * (n + 1) and
A4: seq2 . 0 = 1 and
A5: for n being Element of NAT holds seq2 . (n + 1) = (seq2 . n) * (n + 1) ; :: thesis: seq1 = seq2
defpred S₁[ Element of NAT ] means seq1 . $1 = seq2 . $1;
A6: S₁[ 0 ] by A2, A4;
A7: 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 A8: seq1 . k = seq2 . k ; :: thesis: S₁[k + 1]
thus seq1 . (k + 1) = (seq2 . k) * (k + 1) by A3, A8
.= seq2 . (k + 1) by A5 ; :: thesis: verum

end;

A9: for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A6, A7);
thus seq1 = seq2 by A9, FUNCT_2:113; :: thesis: verum