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
  for a be non trivial Nat, b, n be non zero Nat holds
    b < a implies a |-count b|^n < n
proof
  let a be non trivial Nat, b,n be non zero Nat;
  assume
  A1: b < a;
  b >= 0 + 1 by NAT_1:13; then
  A2: a |-count b = 0 by A1,NAT_3:23;
  a |-count b|^n < n*((a |-count b)+1) by Count2;
  hence thesis by A2;
end;
