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 Th22:
  for n being non zero Nat st n > 1 holds
  support Euler_factorization n is non empty
  proof
    let n be non zero Nat such that
A1: n > 1;
    set f = Euler_factorization n;
    n >= 1+1 by A1,NAT_1:13;
    then consider p being Element of NAT such that
A2: p is prime and
A3: p divides n by INT_2:31;
    dom f = support ppf n by Def1
    .= support pfexp n by NAT_3:def 9;
    then consider c being non zero Nat such that
A4: c = p |-count n & f.p = p|^c - p|^(c-1) by A2,A3,NAT_3:37,Def1;
    c-1 < c-0 by XREAL_1:15;
    then f.p <> 0 by A2,A4,NAT_6:2;
    hence thesis by PRE_POLY:def 7;
  end;
