reserve i,s,t,m,n,k for Nat,
        c,d,e for Element of NAT,
        fn for FinSequence of NAT,
        x,y for Integer;
reserve p for Prime;
 reserve fp,fr for FinSequence of NAT;

theorem
  for X be finite Subset of INT,a be Integer st (m>1 &
    a,m are_coprime & X is_RRS_of m) holds (a ** X) is_RRS_of m
proof let X be finite Subset of INT,a be Integer;
  assume A1:m>1 & a,m are_coprime & X is_RRS_of m;
  then A2:card X = Euler m & (for x,y be Integer st x in X & y in X & x<>y
          holds not x,y are_congruent_mod m) & (for x be Integer st x in X
          holds x,m are_coprime) by Th24;
A3: a ** X c= INT
    by INT_1:def 2;
    a<>0 by A1,Th5;then
A4: card(a ** X) = Euler m by A2,INT_4:5;
A5:   for x,y be Integer st x in (a ** X) & y in (a ** X) & x<>y holds
         not x,y are_congruent_mod m
    proof let x,y be Integer;
      assume A6:x in (a ** X) & y in (a ** X) & x <> y;
      then consider z1 be Integer such that A7:z1 in X & x=a*z1 by Th3;
      consider z2 be Integer such that A8:z2 in X & y=a*z2 by A6,Th3;
      not z1,z2 are_congruent_mod m by A7,A6,A8,A1,Th24;
      hence thesis by A7,A8,A1,INT_4:9;
   end;
   for x be Integer st x in (a ** X) holds x,m are_coprime
   proof let x be Integer;
     assume x in (a ** X);
     then consider y such that A9:y in X & x = a * y by Th3;
     y,m are_coprime by A9,A1,Th24;
     hence thesis by A9, A1,INT_2:26;
   end;
  hence thesis by A1,A3,A4,A5,Th26;
end;
