let f, g be Function of RAT,NAT; :: thesis: ( ( for x being Rational holds f . x = denominator x ) & ( for x being Rational holds g . x = denominator x ) implies f = g )

assume that

A3: for x being Rational holds f . x = denominator x and

A4: for x being Rational holds g . x = denominator x ; :: thesis: f = g

assume that

A3: for x being Rational holds f . x = denominator x and

A4: for x being Rational holds g . x = denominator x ; :: thesis: f = g

now :: thesis: for x being Element of RAT holds f . x = g . x

hence
f = g
; :: thesis: verumlet x be Element of RAT ; :: thesis: f . x = g . x

f . x = denominator x by A3;

hence f . x = g . x by A4; :: thesis: verum

end;f . x = denominator x by A3;

hence f . x = g . x by A4; :: thesis: verum