let s1, s2 be sequence of NAT; :: thesis: ( ( for n being Nat holds s1 . n = n + k ) & ( for n being Nat holds s2 . n = n + k ) implies s1 = s2 )

assume that

A2: for n being Nat holds s1 . n = n + k and

A3: for n being Nat holds s2 . n = n + k ; :: thesis: s1 = s2

let n be Element of NAT ; :: according to FUNCT_2:def 8 :: thesis: s1 . n = s2 . n

( s1 . n = n + k & s2 . n = n + k ) by A2, A3;

hence s1 . n = s2 . n ; :: thesis: verum

assume that

A2: for n being Nat holds s1 . n = n + k and

A3: for n being Nat holds s2 . n = n + k ; :: thesis: s1 = s2

let n be Element of NAT ; :: according to FUNCT_2:def 8 :: thesis: s1 . n = s2 . n

( s1 . n = n + k & s2 . n = n + k ) by A2, A3;

hence s1 . n = s2 . n ; :: thesis: verum