let A, B be Subset of X; :: thesis: ( ( for x being Element of X holds
( x in A iff f . x <> 0. S ) ) & ( for x being Element of X holds
( x in B iff f . x <> 0. S ) ) implies A = B )
assume that
A2: for x being Element of X holds
( x in A iff f . x <> 0. S ) and
A3: for x being Element of X holds
( x in B iff f . x <> 0. S ) ; :: thesis: A = B

now

let x be Element of X; :: thesis: ( x in A iff x in B )
( x in A iff f . x <> 0. S ) by A2;
hence ( x in A iff x in B ) by A3; :: thesis: verum

end;

hence A = B by SUBSET_1:3; :: thesis: verum