reserve a,b,c,d,x,j,k,l,m,n,o,xi,xj for Nat,
  p,q,t,z,u,v for Integer,
  a1,b1,c1,d1 for Complex;

theorem CountD1:
  for b be non zero Integer, a be Integer st |.a.| <> 1 holds
  a|^n divides b & not a|^(n+1) divides b implies
  n = |.a.| |-count |.b.|
proof
  let b be non zero Integer, a be Integer such that
  A0: |.a.| <> 1;
  reconsider k = |.a.|, l = |.b.| as Nat;
  A1: |.a|^n.| = |.a.||^n & |.a|^(n+1).| = |.a.||^(n+1) by TAYLOR_2:1;
  assume a|^n divides b & not a|^(n+1) divides b; then
  |.a.||^n divides |.b.| & not |.a.||^(n+1) divides |.b.| by A1,INT_2:16;
  hence thesis by A0,NAT_3:def 7;
end;
