let s1, s2 be sequence of X; :: thesis: ( ( for n being Nat holds s1 . n = s . (n + k) ) & ( for n being Nat holds s2 . n = s . (n + k) ) implies s1 = s2 )
assume that
A2: for n being Nat holds s1 . n = s . (n + k) and
A3: for n being Nat holds s2 . n = s . (n + k) ; :: thesis: s1 = s2

now

let n be Element of NAT ; :: thesis: s1 . n = s2 . n
s1 . n = s . (n + k) by A2;
hence s1 . n = s2 . n by A3; :: thesis: verum

end;

hence s1 = s2 by FUNCT_2:113; :: thesis: verum