reserve I for set;

theorem Th1:
  for A being set, B, C being non empty set for f being Function of
  A, B, g being Function of B, C st rng (g * f) = C holds rng g = C
proof
  let A be set, B, C be non empty set, f be Function of A, B, g be Function of
  B, C such that
A1: rng (g * f) = C;
  thus rng g c= C;
  let q be object;
  assume q in C;
  then consider x being object such that
A2: x in dom (g * f) and
A3: q = (g * f).x by A1,FUNCT_1:def 3;
A4: dom f = A by FUNCT_2:def 1;
  then
A5: f.x in rng f by A2,FUNCT_1:def 3;
  dom (g * f) = A by FUNCT_2:def 1;
  then
A6: rng f c= dom g by A4,FUNCT_1:15;
  q = g.(f.x) by A2,A3,FUNCT_1:12;
  hence thesis by A6,A5,FUNCT_1:def 3;
end;
