reserve X,Y for set;
reserve G for Group;
reserve n for Nat;

theorem
  for x being Element of SymGroup(X) holds x" = x qua Function"
  proof
    let x be Element of SymGroup(X);
    reconsider f = x as Permutation of X by Th5;
    f" in permutations(X);
    then reconsider g = f" as Element of SymGroup(X) by Def2;
A1: 1_SymGroup(X) = id X by Th6;
    x * g = g qua Function * x by Def2;
    then
A2: x*g = id X by FUNCT_2:61;
    g * x = x qua Function * g by Def2;
    then g*x = id X by FUNCT_2:61;
    hence thesis by A2,A1,GROUP_1:def 5;
  end;
