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 Th81:
  n > 0 & n is having_at_least_three_different_prime_divisors implies n >= 30
  proof
    assume
A1: n > 0;
    given p1,p2,p3 being Prime such that
A2: p1,p2,p3 are_mutually_distinct and
A3: p1 divides n & p2 divides n & p3 divides n;
    p1 >= 2 & p2 >= 3 & p3 >= 5 or p1 >= 2 & p2 >= 5 & p3 >= 3 or
    p1 >= 3 & p2 >= 2 & p3 >= 5 or p1 >= 3 & p2 >= 5 & p3 >= 2 or
    p1 >= 5 & p2 >= 2 & p3 >= 3 or p1 >= 5 & p2 >= 3 & p3 >= 2 by A2,Th49;
    then
A4: p1*p2*p3 >= 2*3*5 or p1*p2*p3 >= 3*5*2 or p1*p2*p3 >= 5*2*3 by Lm15;
A5: p1*p2 divides n by A2,A3,PEPIN:4,INT_2:30;
    p1,p3 are_coprime & p2,p3 are_coprime by A2,INT_2:30;
    then
    p1*p2,p3 are_coprime by EULER_1:14;
    then p1*p2*p3 <= n by A1,A3,A5,NAT_D:7,PEPIN:4;
    hence thesis by A4,XXREAL_0:2;
  end;
