theorem
  f1 is_continuous_on X & f1"{0} = {} & f2 is_continuous_on X implies f2
  /f1 is_continuous_on X
proof
  assume that
A1: f1 is_continuous_on X & f1"{0} = {} and
A2: f2 is_continuous_on X;
  f1^ is_continuous_on X by A1,Th47;
  then f2(#)(f1^) is_continuous_on X by A2,Th43;
  hence thesis by CFUNCT_1:39;
end;
