reserve A, B, X, Y for set;

theorem Th8:
  <:pr2(X,Y),pr1(X,Y):>" = <:pr2(Y,X),pr1(Y,X):>
proof
  set f = <:pr2(X,Y),pr1(X,Y):>, g = <:pr2(Y,X),pr1(Y,X):>;
A1: dom (f") = rng f by FUNCT_1:32
    .= [:Y,X:] by Th4;
A2: now
    let x be object;
    assume x in [:Y,X:];
    then consider x1, x2 being object such that
A3: x1 in Y & x2 in X and
A4: x = [x1,x2] by ZFMISC_1:def 2;
A5: g.(x1,x2) = [x2,x1] & f.(x2,x1) = [x1,x2] by A3,Lm1;
    dom f = [:X,Y:] by Th4;
    then [x2,x1] in dom f by A3,ZFMISC_1:87;
    hence f".x = g.x by A4,A5,FUNCT_1:32;
  end;
  dom g = [:Y,X:] by Th4;
  hence thesis by A1,A2;
end;
