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;
reserve e for object, X,X1,X2,Y1,Y2 for set;

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;
