reserve A for set;
reserve X,Y,Z for set,x,x1,x2,y,y1,y2,z,z1,z2 for object;

theorem Th1:
  for X,Y,Z being set for f1,f2 being Function of [:X,Y:],Z st for
  x,y being set st x in X & y in Y holds f1.(x,y) = f2.(x,y) holds f1 = f2
proof
  let X,Y,Z be set;
  let f1,f2 be Function of [:X,Y:],Z such that
A1: for x,y being set st x in X & y in Y holds f1.(x,y) = f2.(x,y);
  for z being object st z in [:X,Y:] holds f1.z = f2.z
  proof
    let z be object;
    assume z in [:X,Y:];
    then consider x,y being object such that
A2: x in X & y in Y and
A3: z = [x,y] by ZFMISC_1:def 2;
    f1.(x,y) = f1.z & f2.(x,y) = f2.z by A3;
    hence thesis by A1,A2;
  end;
  hence thesis by FUNCT_2:12;
end;
