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
  for f1,f2 being Function of [:A,B,C:],D st for a being Element of A
for b being Element of B for c being Element of C holds f1.(a,b,c) = f2.(a,b,c)
  holds f1 = f2
proof
  let f1,f2 be Function of [:A,B,C:],D such that
A1: for a being Element of A for b being Element of B for c being
  Element of C holds f1.(a,b,c) = f2.(a,b,c);
  for a being Element of A for b being Element of B for c being Element of
  C holds f1.[a,b,c] = f2.[a,b,c]
  proof
    let a be Element of A;
    let b be Element of B;
    let c be Element of C;
    f1.(a,b,c) = f1.[a,b,c] & f2.(a,b,c) = f2.[a,b,c];
    hence thesis by A1;
  end;
  hence thesis by Th2;
end;
