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
  {} is_RRS_of m iff m = 0
proof thus {} is_RRS_of m implies m = 0
   proof assume {} is_RRS_of m;
     then Euler m = card {} by Th24;
     hence thesis by PEPIN:42;
   end;
   assume A1:m = 0;
   reconsider fp=<*>INT as FinSequence of INT;
    RelPrimes m c= Seg m by Th1; then
    card RelPrimes(m) <= card Seg m by NAT_1:43;
    then card RelPrimes(m) <= 0 by A1;
    then card RelPrimes(m) = 0;
    then A2:len Sgm RelPrimes(m) = len fp by Lm3,FINSEQ_3:43;
  (for d st d in dom fp holds fp.d in Class(Cong(m),(Sgm RelPrimes(m)).d));
  hence thesis by RELAT_1:38,A2;
end;
