
theorem Th15:
  for X,Y be non empty set,
      T be Function of X,Y, f be PartFunc of X,ExtREAL,
      g be PartFunc of Y,ExtREAL st T is bijective & g = f*T" holds
    dom g = T.:(dom f) & dom g = (.:T).(dom f)
proof
    let X,Y be non empty set, T be Function of X,Y,
    f be PartFunc of X,ExtREAL, g be PartFunc of Y,ExtREAL;
    assume
A1: T is bijective & g = f*T"; then
    dom g = (T")"(dom f) by RELAT_1:147;
    hence dom g = T.:(dom f) by A1,FUNCT_1:84;
    hence dom g = (.:T).(dom f) by A1,Th1;
end;
