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
  for A,B,C being set st Funcs(A,B) <> {} & Funcs(B,C) <> {} holds
  Funcs(A,C) <> {}
proof
  let A,B,C be set such that
A1: Funcs(A,B) <> {} and
A2: Funcs(B,C) <> {};
  consider g being object such that
A3: g in Funcs(B,C) by A2,XBOOLE_0:def 1;
  ex f being object st f in Funcs(A,B) by A1,XBOOLE_0:def 1;
  hence thesis by A3,Th127;
end;
