
theorem Th17: :: Center1:
  for R being Ring, y being Element of R
  holds y in center R iff for s being Element of R holds y*s = s*y
proof
  let R be Ring, y be Element of R;
  hereby
    assume y in center R;
    then y in the carrier of center R;
    then y in {x where x is Element of R:
    for s being Element of R holds x*s = s*x} by Def4;
    then ex x being Element of R st ( x = y)&( for s being Element
    of R holds x*s=s*x);
    hence for s being Element of R holds y*s = s*y;
  end;
  now
    assume for s being Element of R holds y*s = s*y;
    then y in {x where x is Element of R:
    for s being Element of R holds x*s = s*x};
    then y is Element of center R by Def4;
    hence y in center R;
  end;
  hence thesis;
end;
