theorem Th18:
  for x, y being Element of InnAutGroup G
  for f, g being Element of InnAut G st
    x = f & y = g holds x * y = f * g
proof
  let x,y be Element of InnAutGroup G;
  let f,g be Element of InnAut G;
  x is Element of AutGroup G & y is Element of AutGroup G by GROUP_2:42;
  then consider x1, y1 be Element of AutGroup G such that
A1: x1 = x & y1 = y;
  f is Element of Aut G & g is Element of Aut G by Th12;
  then consider f1, g1 be Element of Aut G such that
A2: f1 = f & g1 = g;
  assume x = f & y = g;
  then x1 * y1 = f1 * g1 by A2,A1,Def2;
  hence thesis by A2,A1,GROUP_2:43;
end;
