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
  for G being strict Group
  for f being object
  holds f in InnAut G iff f is inner Automorphism of G
proof
  let G be strict Group;
  let f be object;
  A1: f is Automorphism of G implies
      f is Element of Funcs (the carrier of G, the carrier of G) by FUNCT_2:9;
  thus f in InnAut G implies f is inner Automorphism of G
  proof
    assume B1: f in InnAut G;
    then reconsider f as Automorphism of G by Lm6;
    consider a being Element of G such that
    B2: for x being Element of G holds f.x = x |^ a by A1,B1,AUTGROUP:def 4;
    a is_inner_wrt f by B2;
    hence thesis by Def2;
  end;
  assume f is inner Automorphism of G;
  then reconsider f as inner Automorphism of G;
  ex a being Element of G st a is_inner_wrt f by Def2;
  hence thesis by A1,AUTGROUP:def 4;
end;
