
theorem Th2:
  for X,Y being non empty MetrSpace, f being Function of
TopSpaceMetr(X),TopSpaceMetr(Y),S being sequence of X holds f*S is sequence of
  Y
proof
  let X,Y be non empty MetrSpace, f be Function of TopSpaceMetr(X),
  TopSpaceMetr(Y),S be sequence of X;
A1: dom f=the carrier of TopSpaceMetr(X) by FUNCT_2:def 1
    .=the carrier of X by TOPMETR:12;
  dom S=NAT & rng S c= the carrier of X by FUNCT_2:def 1;
  then dom (f*S)=NAT by A1,RELAT_1:27;
  hence thesis by FUNCT_2:def 1,TOPMETR:12;
end;
