 reserve x,y,X,Y for set;
reserve G for non empty multMagma,
  D for set,
  a,b,c,r,l for Element of G;
reserve M for non empty multLoopStr;
reserve H for non empty SubStr of G,
  N for non empty MonoidalSubStr of G;

theorem
  for f being Element of GPerms X holds f" = (f qua Function)"
proof
  let f be Element of GPerms X;
  reconsider g = f as Permutation of X by Th77;
A1: g(*)(g") = id X & g"(*)g = id X by FUNCT_2:61;
  reconsider b = g" as Element of GPerms X by Th77;
  reconsider b as Element of GPerms X;
A2: b[*]f = g(*)(g") by Th70;
  id X = 1_GPerms X & f[*]b = g"(*)g by Th70,Th78;
  hence thesis by A1,A2,GROUP_1:def 5;
end;
