let J1, J2 be SetSequence of X; :: thesis: ( ( for n being Nat holds J1 . n = Union (A ^\ n) ) & ( for n being Nat holds J2 . n = Union (A ^\ n) ) implies J1 = J2 )
assume that
A8: for n being Nat holds J1 . n = Union (A ^\ n) and
A9: for n being Nat holds J2 . n = Union (A ^\ n) ; :: thesis: J1 = J2
for n being Element of NAT holds J1 . n = J2 . n

proof

let n be Element of NAT ; :: thesis: J1 . n = J2 . n
J1 . n = Union (A ^\ n) by A8;
hence J1 . n = J2 . n by A9; :: thesis: verum

end;

then for n being object st n in NAT holds
J1 . n = J2 . n ;
hence J1 = J2 ; :: thesis: verum