reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;

theorem Th18:
  for n being non zero Nat
  for p being Nat st p in dom Euler_factorization n holds
  ex c being non zero Nat st c = p |-count n &
  (Euler_factorization n).p = p|^(c-1) * (p-1)
  proof
    let n be non zero Nat;
    let p be Nat;
    set f = Euler_factorization n;
    assume p in dom f;
    then consider c being non zero Nat such that
A1: c = p |-count n & f.p = p|^c - p|^(c-1) by Def1;
    take c;
    p|^c = p|^(c-1+1)
    .= p|^(c-1)*p by NEWTON:6;
    hence thesis by A1;
  end;
