reserve u,v,x,x1,x2,y,y1,y2,z,p,a for object,
        A,B,X,X1,X2,X3,X4,Y,Y1,Y2,Z,N,M for set;

theorem
  X <> {} & Y <> {} & [:X,Y:] = [:Y,X:] implies X = Y
proof
  assume X<>{};
  then consider x such that
A1: x in X by XBOOLE_0:7;
  assume Y<>{};
  then consider y such that
A2: y in Y by XBOOLE_0:7;
  assume
A3: [:X,Y:]=[:Y,X:];
  for z holds z in X iff z in Y
  proof let z;
    thus z in X implies z in Y
    proof
      assume z in X;
      then [z,y] in [:Y,X:] by A2,A3,Lm16;
      hence thesis by Lm16;
    end;
    assume z in Y;
    then [z,x] in [:X,Y:] by A1,A3,Lm16;
    hence thesis by Lm16;
  end;
  hence thesis by TARSKI:2;
end;
