reserve A, B, X, Y for set;

theorem
  for A being Subset of X, B being Subset of Y holds <:pr2(X,Y),pr1(X,Y)
  :>.:[:A,B:] = [:B,A:]
proof
  let A be Subset of X, B be Subset of Y;
  set f = <:pr2(X,Y),pr1(X,Y):>;
A1: dom f = [:X,Y:] by Th4;
  thus f.:[:A,B:] c= [:B,A:] by Th5;
  let y be object;
  assume y in [:B,A:];
  then consider y1, y2 being object such that
A2: y1 in B & y2 in A and
A3: y = [y1,y2] by ZFMISC_1:def 2;
  [y2,y1] in [:A,B:] & f.(y2,y1) = [y1,y2] by A2,Lm1,ZFMISC_1:87;
  hence thesis by A1,A3,FUNCT_1:def 6;
end;
