reserve X, Y for non empty set;

theorem Th2:
  for R,S being RMembership_Func of X,Y st for x being Element of X
  , y being Element of Y holds R.(x,y) = S.(x,y) holds R = S
proof
  let R,S be RMembership_Func of X,Y;
  assume
A1: for x being Element of X, y being Element of Y holds R.(x,y) = S.(x, y);
A2: for x,y being object st [x,y] in dom R holds R.(x,y) = S.(x,y)
  proof
    let x,y be object;
    assume
A3: [x,y] in dom R;
    then reconsider x as Element of X by ZFMISC_1:87;
    reconsider y as Element of Y by A3,ZFMISC_1:87;
    R.(x,y) = S.(x,y) by A1;
    hence thesis;
  end;
  dom R = [:X,Y:] & dom S = [:X,Y:] by FUNCT_2:def 1;
  hence thesis by A2,BINOP_1:20;
end;
