
theorem Th1:
  for A being Subset of REAL, x being Real st x <> 0 holds
  x" ** (x ** A) = A
proof
  let A be Subset of REAL;
  let x be Real;
  assume
A1: x <> 0;
  thus x" ** (x ** A) c= A
  proof
    let y be object;
    assume
A2: y in x" ** (x ** A);
    consider z being Real such that
A3: z in x ** A and
A4: y = x" * z by A2,INTEGRA2:39;
    consider t being Real such that
A5: t in A and
A6: z = x * t by A3,INTEGRA2:39;
    y = (x" * x) * t by A4,A6
      .= 1 * t by A1,XCMPLX_0:def 7
      .= t;
    hence thesis by A5;
  end;
  let y be object;
  assume
A7: y in A;
  then reconsider y as Real;
  set t = y / x";
A8: t in x ** A by A7,MEMBER_1:193;
  y = x" * t by A1,XCMPLX_1:87,202;
  hence thesis by A8,MEMBER_1:193;
end;
