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

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

let m be Element of NAT ; :: thesis: f1 . m = f2 . m
thus f1 . m = 1 --> m by A3
.= f2 . m by A4 ; :: thesis: verum

end;

hence f1 = f2 ; :: thesis: verum