theorem
  for A being non empty set, B,C,D being set st [:A,B:] c= [:C,D:] or [:
  B,A:] c= [:D,C:] holds B c= D
proof
  let A be non empty set, B,C,D be set such that
A1: [:A,B:] c= [:C,D:] or [:B,A:] c= [:D,C:];
  per cases;
  suppose
    B = {};
    hence thesis;
  end;
  suppose
    B <> {};
    then [:A,B:] <> {} & [:B,A:] <> {} by Th89;
    hence thesis by A1,Th113;
  end;
end;
