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 Th21:
  for n being non zero Nat holds
  support Euler_factorization n = dom Euler_factorization n
  proof
    let n be non zero Nat;
    set f = Euler_factorization n;
    thus support f c= dom f by PRE_POLY:37;
    let p be object;
    assume
A1: p in dom f;
    then reconsider p as Prime by Th17;
    consider c being non zero Nat such that
A2: c = p |-count n & f.p = p|^c - p|^(c-1) by A1,Def1;
    c-1 < c-0 by XREAL_1:15;
    then f.p <> 0 by A2,NAT_6:2;
    hence thesis by PRE_POLY:def 7;
  end;
