reserve A,B,C,D,E for non empty set,
  a for Element of A,
  b for Element of B,
  c for Element of C,
  d for Element of D,
  X,Y,Z,S for set,x,y,z,s,t for object;

theorem Th4:
  for f1,f2 being Function of [:X,Y,Z,S:],D st for x,y,z,s st x in
  X & y in Y & z in Z & s in S holds f1.[x,y,z,s] = f2.[x,y,z,s] holds f1 = f2
proof
  let f1,f2 be Function of [:X,Y,Z,S:],D such that
A1: for x,y,z,s st x in X & y in Y & z in Z & s in S holds f1.[x,y,z,s]
  = f2.[x,y,z,s];
  for t being object st t in [:X,Y,Z,S:] holds f1.t = f2.t
  proof
    let t be object;
    assume t in [:X,Y,Z,S:];
    then ex x,y,z,s st x in X & y in Y & z in Z & s in S & t = [x,y,z,s] by
MCART_1:79;
    hence thesis by A1;
  end;
  hence thesis by FUNCT_2:12;
end;
