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 Th22:
  RelPrimes(m) is_RRS_of m
proof
A2: rng Sgm RelPrimes(m) = RelPrimes(m) &
    len Sgm RelPrimes(m) = len Sgm RelPrimes(m) by FINSEQ_1:def 14;
    rng Sgm RelPrimes(m) c= INT by RELAT_1:def 19;
    then A3:Sgm RelPrimes(m) is FinSequence of INT by FINSEQ_1:def 4;
    for a be Nat st a in dom Sgm RelPrimes(m)
      holds (Sgm RelPrimes(m)).a in Class(Cong(m),(Sgm RelPrimes(m)).a)
    proof let a be Nat;
      assume a in dom Sgm RelPrimes(m);
      (Sgm RelPrimes(m)).a,(Sgm RelPrimes(m)).a are_congruent_mod m
        by INT_1:11;
      then [(Sgm RelPrimes(m)).a,(Sgm RelPrimes(m)).a] in Cong(m)
        by INT_4:def 1;
      hence thesis by EQREL_1:18;
    end; then
    for d st d in dom Sgm RelPrimes(m) holds (Sgm RelPrimes(m)).d
      in Class(Cong m,(Sgm RelPrimes(m)).d);
    hence thesis by A2,A3;
end;
