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 Th85:
  m divides p|^4 iff m in {1,p,p|^2,p|^3,p|^4}
  proof
    thus m divides p|^4 implies m in {1,p,p|^2,p|^3,p|^4}
    proof
      assume m divides p|^4;
      then consider r being Nat such that
A1:   m = p|^r and
A2:   r <= 4 by GROUPP_1:2;
      r = 0 or ... or r = 4 by A2;
      then m = 1 or m = p or m = p|^2 or m = p|^3 or m = p|^4 by A1,NEWTON:4;
      hence thesis by ENUMSET1:def 3;
    end;
    assume m in {1,p,p|^2,p|^3,p|^4};
    then m = 1 or m = p|^1 or m = p|^2 or m = p|^3 or m = p|^4
    by ENUMSET1:def 3;
    hence thesis by INT_2:12,NEWTON:89;
  end;
