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 Count1:
  for a be non trivial Nat, b be non zero Integer holds
    a|^n divides b iff n <= a |-count b
proof
  let a be non trivial Nat, b be non zero Integer;
  L0: a <> 1 by Def0;
  L1: a|^n divides b implies n <= a |-count b
  proof
    assume
    A1: a|^n divides b;
    a|^n divides b & not a|^((a |-count b)+1) divides b implies
      not a|^((a |-count b)+1) divides a|^n by INT_2:9; then
    (a |-count b) + 1 > n by L0,A1,Def6, NEWTON:89;
    hence thesis by NAT_1:13;
  end;
  not a|^(n) divides b implies a |-count b < n
  proof
    assume
    A1: not a|^n divides b;
    not a|^n divides b & a|^(a |-count b) divides b implies
      not a|^n divides a|^(a |-count b) by INT_2:9;
    hence thesis by L0,A1,Def6,NEWTON:89;
  end;
  hence thesis by L1;
end;
