
theorem LmEqRng:
  for X,Y,Z,W being set st Z <> {} & W <> {}
  for f being Function of [:X,Y:],Z
  for g being Function of [:X,Y:],W
  st (for a being Element of X, b being Element of Y holds f.(a,b) = g.(a,b))
  holds f = g
proof
  let X,Y,Z,W be set;
  assume A0: Z <> {} & W <> {};
  let f be Function of [:X,Y:],Z;
  let g be Function of [:X,Y:],W;
  assume A1: for a being Element of X, b being Element of Y
  holds f.(a,b) = g.(a,b);
A2: dom f = [:X,Y:] & dom g = [:X,Y:] by FUNCT_2:def 1,A0;
  for x,y being object st x in X & y in Y holds f.(x,y) = g.(x,y) by A1;
  hence thesis by A2,FUNCT_3:6;
end;
