reserve UA for Universal_Algebra,
  f, g for Function of UA, UA;

theorem Th8:
  id the carrier of UA = 1_UAAutGroup UA
proof
  set f = the Element of UAAutGroup UA;
  reconsider g = id the carrier of UA as Element of UAAutGroup UA by Th3;
  consider g1 be Function of the carrier of UA, the carrier of UA such that
A1: g1 = g;
  f is Element of UAAut UA;
  then consider
  f1 be Function of the carrier of UA, the carrier of UA such that
A2: f1 = f;
  g * f = f1 * g1 by A1,A2,Def2
    .= f by A1,A2,FUNCT_2:17;
  hence thesis by GROUP_1:7;
end;
