
theorem Th2:
  for x being Real st x <> 0 holds for A being Subset of REAL holds
  A = REAL implies x ** A = A
proof
  let x be Real;
  assume
A1: x <> 0;
  let A be Subset of REAL;
  assume
A2: A = REAL;
  for y being object st y in A holds y in x ** A
  proof
    let y be object;
    assume y in A;
    then reconsider y as Real;
    reconsider z = y/x as Element of REAL by XREAL_0:def 1;
    y = x * z by A1,XCMPLX_1:87;
    hence thesis by A2,MEMBER_1:193;
  end;
  then A c= x ** A;
  hence thesis by A2;
end;
