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;

theorem Th32:
  for a being Element of G
   ex f being inner Automorphism of G st a is_inner_wrt f
proof
  let a be Element of G;
  deffunc F(Element of G) = ($1) |^ a;
  consider f be Function of the carrier of G, the carrier of G such that
A1: for g being Element of G holds f.g = F(g) from FUNCT_2:sch 4;
  for x1,x2 being Element of G holds f.(x1 * x2) = f.x1 * f.x2
  proof
    let x1,x2 be Element of G;
    A2: f.x1 = x1 |^ a & f.x2 = x2 |^ a by A1;
    f.(x1 * x2) = (x1 * x2) |^ a by A1
               .= (x1 |^ a) * (x2 |^ a) by GROUP_3:23
               .= f.x1 * f.x2 by A2;
    hence thesis;
  end;
  then reconsider f as Endomorphism of G by GROUP_6:def 6;
  a is_inner_wrt f & f is inner Automorphism of G by A1,Th30,Def2,Def2a;
  hence thesis;
end;
