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

J1 . n = J2 . n ;

hence J1 = J2 ; :: thesis: verum

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

then
for n being object st n in NAT holds
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;J1 . n = Union (A ^\ n) by A8;

hence J1 . n = J2 . n by A9; :: thesis: verum

J1 . n = J2 . n ;

hence J1 = J2 ; :: thesis: verum