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 Th57:
  for G being Group
  for A being Subset of G
  for g being Element of G
  holds (for a being Element of G st a in A holds g*a = a*g) iff
        g is Element of Centralizer A
proof
  let G be Group;
  let A be Subset of G;
  let g be Element of G;
A1: the carrier of Centralizer A = {b where b is Element of G : for a
  being Element of G st a in A holds b*a=a*b} by Def4;
  hereby
    assume for a being Element of G st a in A holds g*a = a*g;
    then g in the carrier of Centralizer A by A1;
    hence g is Element of Centralizer A;
  end;
  assume g is Element of Centralizer A;
  then g in the carrier of Centralizer A;
  then ex b being Element of G st (b = g) & (for a being Element of G st a in A
  holds b*a = a*b) by A1;
  hence thesis;
end;
