let f, g be Function of (Ideals (A / q)),(Ideals (A,q)); :: thesis: ( ( for x being Element of Ideals (A / q) holds f . x = (canHom q) " x ) & ( for x being Element of Ideals (A / q) holds g . x = (canHom q) " x ) implies f = g )
assume that
A7: for x being Element of Ideals (A / q) holds f . x = (canHom q) " x and
A8: for x being Element of Ideals (A / q) holds g . x = (canHom q) " x ; :: thesis: f = g

now :: thesis: for x being Element of Ideals (A / q) holds f . x = g . x

let x be Element of Ideals (A / q); :: thesis: f . x = g . x
f . x = (canHom q) " x by A7;
hence f . x = g . x by A8; :: thesis: verum

end;

hence f = g ; :: thesis: verum