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 Th1:
  for f1,f2 being Function of [:X,Y,Z:],D st for x,y,z st x in X &
  y in Y & z in Z holds f1.[x,y,z] = f2.[x,y,z] holds f1 = f2
proof
  let f1,f2 be Function of [:X,Y,Z:],D such that
A1: for x,y,z st x in X & y in Y & z in Z holds f1.[x,y,z] = f2.[x,y,z];
  for t being object st t in [:X,Y,Z:] holds f1.t = f2.t
  proof
    let t be object;
    assume t in [:X,Y,Z:];
    then ex x,y,z st x in X & y in Y & z in Z & t = [x,y,z] by MCART_1:68;
    hence thesis by A1;
  end;
  hence thesis by FUNCT_2:12;
end;
