
theorem Th7:
  for G being Group, a being Element of G, x being set
  st x in Centralizer a holds x in G
proof
  let G be Group, a be Element of G, x be set;
  assume
A1: x in Centralizer a;
  the carrier of Centralizer a =
  { b where b is Element of G : a*b = b*a } by Def1;
  then x in { b where b is Element of G : a*b = b*a } by A1;
  then ex b being Element of G st b = x & a*b = b*a;
  hence thesis;
end;
