reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;
reserve D for non empty set;
reserve A,B for non empty set;
reserve Y for non empty set,
  f for Function of X,Y,
  p for PartFunc of Y,Z,
  x for Element of X;
reserve g for Function of X,X;
reserve X,Y for non empty set,
  Z,S,T for set,
  f for Function of X,Y,
  g for PartFunc of Y,Z,
  x for Element of X;

theorem Th127:
  for A,B,C being set, f,g being Function st f in Funcs(A,B) & g in
  Funcs(B,C) holds g*f in Funcs(A,C)
proof
  let A,B,C be set, f,g be Function;
  assume that
A1: f in Funcs(A,B) and
A2: g in Funcs(B,C);
A3: ex g9 being Function st g9 = g & dom g9 = B & rng g9 c= C by A2,Def2;
  rng(g*f) c= rng g by RELAT_1:26;
  then
A4: rng(g*f) c= C by A3;
  ex f9 being Function st f9 = f & dom f9 = A & rng f9 c= B by A1,Def2;
  then dom(g*f) = A by A3,RELAT_1:27;
  hence thesis by A4,Def2;
end;
