let f1, f2 be Function of (Bags 1),NAT; :: thesis: ( ( for b being Element of Bags 1 holds f1 . b = b . 0 ) & ( for b being Element of Bags 1 holds f2 . b = b . 0 ) implies f1 = f2 )
assume that
A3: for m being Element of Bags 1 holds f1 . m = m . 0 and
A4: for m being Element of Bags 1 holds f2 . m = m . 0 ; :: thesis: f1 = f2

now :: thesis: for m being Element of Bags 1 holds f1 . m = f2 . m

let m be Element of Bags 1; :: thesis: f1 . m = f2 . m
thus f1 . m = m . 0 by A3
.= f2 . m by A4 ; :: thesis: verum

end;

hence f1 = f2 ; :: thesis: verum