let s1, s2 be Complex_Sequence; :: thesis: ( s1 . 0 = s . 0 & ( for n being Nat holds s1 . (n + 1) = (s1 . n) * (s . (n + 1)) ) & s2 . 0 = s . 0 & ( for n being Nat holds s2 . (n + 1) = (s2 . n) * (s . (n + 1)) ) implies s1 = s2 )
assume A1: ( s1 . 0 = s . 0 & ( for n being Nat holds s1 . (n + 1) = (s1 . n) * (s . (n + 1)) ) & s2 . 0 = s . 0 & ( for n being Nat holds s2 . (n + 1) = (s2 . n) * (s . (n + 1)) ) ) ; :: thesis: s1 = s2
for n being Element of NAT holds s1 . n = s2 . n

proof

defpred S₁[ Nat] means s1 . $1 = s2 . $1;
B1: S₁[ 0 ] by A1;
B2: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume s1 . n = s2 . n ; :: thesis: S₁[n + 1]
then s1 . (n + 1) = (s2 . n) * (s . (n + 1)) by A1
.= s2 . (n + 1) by A1 ;
hence S₁[n + 1] ; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(B1, B2);
hence for n being Element of NAT holds s1 . n = s2 . n ; :: thesis: verum

end;

hence s1 = s2 by FUNCT_2:63; :: thesis: verum