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 Th45:
  for n being non zero Nat holds Euler_factorization_2 n <= n-1
  proof
    let n be non zero Nat;
    set f = Euler_factorization_2 n;
    let x be object such that
A1: x in dom f;
    reconsider p = x as Prime by A1,Th29;
A2: f.p = p-1 by A1,Def3;
A3: support ppf n = support pfexp n by NAT_3:def 9;
    dom f = support ppf n by Def3;
    then p <= n by A1,A3,NAT_3:36,NAT_D:7;
    hence f.x <= n-1 by A2,XREAL_1:7;
  end;
