theorem Th11:
  1 * AR = AR
  proof
    hereby let x be object;
      assume x in 1*AR;
      then ex v be Element of R st x=1*v & v in AR;
      hence x in AR by RLVECT_1:def 8;
    end;
    let x be object such that
    A1: x in AR;
    reconsider v=x as Element of R by A1;
    x=1*v by RLVECT_1:def 8;
    hence thesis by A1;
  end;
