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
  131 satisfies_Sierpinski_problem_105
  proof
A1: 130 = 2*5*13;
A2: 2,5,13 are_mutually_distinct;
    130 = 5*13*2 & 130 = 13*2*5;
    then 2 divides 130 & 5 divides 130 & 13 divides 130 by A1;
    hence 131-1 is having_at_least_three_different_prime_divisors
    by A2,XPRIMES1:2,5,13;
A3: 132 = 2*2*3*11;
A4: 2,3,11 are_mutually_distinct;
    132 = 3*11*2*2 & 132 = 11*2*2*3;
    then 2 divides 132 & 3 divides 132 & 11 divides 132 by A3;
    hence 131+1 is having_at_least_three_different_prime_divisors
    by A4,XPRIMES1:2,3,11;
  end;
