
theorem Th3:
  for A being Subset of REAL st A <> {} holds 0 ** A = {0}
proof
  let A be Subset of REAL;
  assume
A1: A <> {};
A2: {0} c= 0 ** A
  proof
    let y be object;
    consider t being object such that
A3: t in A by A1,XBOOLE_0:def 1;
    reconsider t as Element of A by A3;
    reconsider t as Real;
    assume y in {0};
    then y = 0 * t by TARSKI:def 1;
    hence thesis by A3,MEMBER_1:193;
  end;
  0 ** A c= {0}
  proof
    let y be object;
    assume
A4: y in 0 ** A;
    then reconsider y as Real;
    ex z being Real st z in A & y = 0 * z by A4,INTEGRA2:39;
    hence thesis by TARSKI:def 1;
  end;
  hence thesis by A2;
end;
