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,b be non trivial Nat, c be non zero Nat holds
    (a |-count b)*(b|-count c) <= (a |-count c)
  proof
    let a,b be non trivial Nat, c be non zero Nat;
    a <> 1 & b <> 1 by Def0; then
A2: a|^(a |-count b) divides b & b|^(b |-count c) divides c &
      a|^(a |-count c) divides c
    & not a|^((a |-count c)+1) divides c by NAT_3:def 7; then
    (a|^(a |-count b))|^(b |-count c) divides c by LmY; then
    a|^((a |-count b)*(b |-count c)) divides c by NEWTON:9; then
    ((a |-count b)*(b |-count c)) < (a |-count c)+1 by A2,LmX;
    hence thesis by NAT_1:13;
  end;
