reserve E, x, y, X for set;
reserve A, B, C for Subset of E^omega;
reserve a, b for Element of E^omega;
reserve i, k, l, kl, m, n, mn for Nat;

theorem
  <%>E in A & <%>E in B implies A? c= A ^^ B & A? c= B ^^ A
proof
  assume that
A1: <%>E in A and
A2: <%>E in B;
  <%>E in B ^^ A by A1,A2,FLANG_1:15;
  then
A3: {<%>E} c= B ^^ A by ZFMISC_1:31;
  <%>E in A ^^ B by A1,A2,FLANG_1:15;
  then
A4: {<%>E} c= A ^^ B by ZFMISC_1:31;
  A c= B ^^ A by A2,FLANG_1:16;
  then
A5: {<%>E} \/ A c= B ^^ A by A3,XBOOLE_1:8;
  A c= A ^^ B by A2,FLANG_1:16;
  then {<%>E} \/ A c= A ^^ B by A4,XBOOLE_1:8;
  hence thesis by A5,Th76;
end;
