reserve a,b,m,x,y,i1,i2,i3,i for Integer,
  k,p,q,n for Nat,
  c,c1,c2 for Element of NAT,
  z for set;
reserve fp,fp1 for FinSequence of NAT,

  b,c,d, n for Element of NAT,
  a for Nat;
reserve i,m,m1,m2,m3,r,s,a,b,c,c1,c2,x,y for Integer;
reserve

  a,b,c,m for Element of NAT;

theorem Th50:
  {} is_CRS_of m iff m = 0
proof
  set fp=<*>INT;
  thus {} is_CRS_of m implies m = 0 by Th49,CARD_1:27;
  assume m = 0;
  then
A1: len fp = m;
  {} = rng fp & for b be Nat st b in dom fp holds fp.b in Class(Cong(m),b -'1);
  hence thesis by A1;
end;
