reserve X for set;
reserve G for Group;
reserve H for Subgroup of G;
reserve h,x,y for object;
reserve f for Endomorphism of G;
reserve phi for Automorphism of G;
reserve K for characteristic Subgroup of G;

theorem Th60:
  for G being Group
  for H being Subgroup of G
  for g being Element of G
  holds (for a being Element of G st a in H holds g*a = a*g) iff
        g is Element of Centralizer H
proof
  let G be Group;
  let H be Subgroup of G;
  let g be Element of G;
A1: the carrier of Centralizer H = {b where b is Element of G : for a
  being Element of G st a in H holds b*a=a*b} by Th59;
  hereby
    assume for a being Element of G st a in H holds g*a = a*g;
    then g in the carrier of Centralizer H by A1;
    hence g is Element of Centralizer H;
  end;
  assume g is Element of Centralizer H;
  then g in the carrier of Centralizer H;
  then ex b being Element of G st (b = g) & (for a being Element of G st a in H
  holds b*a = a*b) by A1;
  hence thesis;
end;
