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 ::: (p|^a)|^c = p|^b implies a divides b  contraposed
  for p be non trivial Nat holds
    not a divides b implies (p|^a)|^c <> p|^b
proof
  let p be non trivial Nat;
  assume
  A1: not a divides b;
  A2: p|^(a*c) = p|^b iff (p|^a)|^c = p|^b by NEWTON:9;
  p > 1 by Def0;
  hence thesis by A1,A2,PEPIN:30;
end;
