let X, Y, Z be non empty TopSpace; :: thesis: ( the carrier of Y = the carrier of Z & the topology of Z c= the topology of Y implies for f being continuous Function of X,Y holds f is continuous Function of X,Z )

assume that

A1: the carrier of Y = the carrier of Z and

A2: the topology of Z c= the topology of Y ; :: thesis: for f being continuous Function of X,Y holds f is continuous Function of X,Z

let f be continuous Function of X,Y; :: thesis: f is continuous Function of X,Z

reconsider g = f as Function of X,Z by A1;

for x being Point of X holds g is_continuous_at x by Th44, A1, A2, Th45;

hence f is continuous Function of X,Z by Th44; :: thesis: verum

assume that

A1: the carrier of Y = the carrier of Z and

A2: the topology of Z c= the topology of Y ; :: thesis: for f being continuous Function of X,Y holds f is continuous Function of X,Z

let f be continuous Function of X,Y; :: thesis: f is continuous Function of X,Z

reconsider g = f as Function of X,Z by A1;

for x being Point of X holds g is_continuous_at x by Th44, A1, A2, Th45;

hence f is continuous Function of X,Z by Th44; :: thesis: verum