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:],E st for a,b,c,d holds f1.(a,b
  ,c,d) = f2.(a,b,c,d) holds f1 = f2
proof
  let f1,f2 be Function of [:A,B,C,D:],E such that
A1: for a,b,c,d holds f1.(a,b,c,d) = f2.(a,b,c,d);
  for a,b,c,d holds f1.[a,b,c,d] = f2.[a,b,c,d]
  proof
    let a,b,c,d;
    f1.(a,b,c,d) = f1.[a,b,c,d] & f2.(a,b,c,d) = f2.[a,b,c,d];
    hence thesis by A1;
  end;
  hence thesis by Th5;
end;
