reserve A for set;
reserve X,Y,Z for set,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve u for UnOp of A,
  o,o9 for BinOp of A,
  a,b,c,e,e1,e2 for Element of A;

theorem
  for f1,f2 being PartFunc of [:X,Y:],Z st dom f1 = dom f2 &
  for x,y being object st
  [x,y] in dom f1 holds f1.(x,y)=f2.(x,y) holds f1 = f2
proof
  let f1,f2 be PartFunc of [:X,Y:],Z such that
A1: dom f1 = dom f2 and
A2: for x,y being object st [x,y] in dom f1 holds f1.(x,y) = f2.(x,y);
  for z being object holds z in dom f1 implies f1.z = f2.z
  proof let z be object;
    assume
A3: z in dom f1;
    then consider x,y being object such that
A4: z = [x,y] by RELAT_1:def 1;
    f1.(x,y) = f2.(x,y) by A2,A3,A4;
    hence thesis by A4;
  end;
  hence thesis by A1;
end;
