
theorem Th8:
  for G being Group, a, x being Element of G
  holds a*x = x*a iff x is Element of Centralizer a
proof
  let G be Group, a, x be Element of G;
A1: the carrier of Centralizer a =
  { b where b is Element of G : a*b = b*a } by Def1;
  hereby
    assume a*x = x*a;
    then x in the carrier of Centralizer a by A1;
    hence x is Element of Centralizer a;
  end;
  assume x is Element of Centralizer a;
  then x in the carrier of Centralizer a;
  then ex b being Element of G st b = x & a*b = b*a by A1;
  hence thesis;
end;
