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

theorem
  SymGroup(Seg n) = Group_of_Perm(n)
  proof
A1: the carrier of SymGroup(Seg n) = permutations(Seg n) by Def2;
A2: permutations(Seg n) = Permutations(n) by Th3
    .= the carrier of Group_of_Perm(n) by MATRIX_1:def 13;
    now
      let a,b be Element of SymGroup(Seg n);
A3:   a * b = b qua Function * a by Def2;
      a is Permutation of Seg n & b is Permutation of Seg n by Th5;
      then a in Permutations(n) & b in Permutations(n) by MATRIX_1:def 12;
      hence (the multF of SymGroup(Seg n)).(a,b) =
      (the multF of Group_of_Perm(n)).(a,b) by A3,MATRIX_1:def 13;
    end;
    hence thesis by A1,A2,BINOP_1:2;
  end;
