reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r for Real;
reserve p,p1,p2 for Prime;

theorem
  not ex n st
  n satisfies_Sierpinski_problem_89 & n+1 satisfies_Sierpinski_problem_89 &
  n+2 satisfies_Sierpinski_problem_89 & n+3 satisfies_Sierpinski_problem_89
  proof
    given n such that
A1: n satisfies_Sierpinski_problem_89 and
    n+1 satisfies_Sierpinski_problem_89 and
    n+2 satisfies_Sierpinski_problem_89 and
A2: n+3 satisfies_Sierpinski_problem_89;
A3: n is_a_product_of_two_different_primes by A1;
A4: n+1 is_a_product_of_two_different_primes by A1;
A5: n+2 is_a_product_of_two_different_primes by A1;
A6: n+3 is_a_product_of_two_different_primes by A2;
    per cases by NUMBER06:19;
    suppose
A7:   4 divides n;
      consider p,q being Prime such that
A8:   p <> q and
A9:   n = p*q by A3;
A10:  p divides p*q & q divides p*q;
      4 divides p*q by A9,A7;
      then p*q = 4 by GR_CY_3:1,XPRIMES0:4;
      then (p = 1 or p = 2) & (q = 1 or q = 2) by A10,NUMBER05:21;
      hence thesis by A8,INT_2:def 4;
    end;
    suppose
A11:  4 divides n+1;
      consider p,q being Prime such that
A12:  p <> q and
A13:  n+1 = p*q by A4;
A14:  p divides p*q & q divides p*q;
      4 divides p*q by A13,A11;
      then p*q = 4 by GR_CY_3:1,XPRIMES0:4;
      then (p = 1 or p = 2) & (q = 1 or q = 2) by A14,NUMBER05:21;
      hence thesis by A12,INT_2:def 4;
    end;
    suppose
A15:  4 divides n+2;
      consider p,q being Prime such that
A16:  p <> q and
A17:  n+2 = p*q by A5;
A18:  p divides p*q & q divides p*q;
      4 divides p*q by A17,A15;
      then p*q = 4 by GR_CY_3:1,XPRIMES0:4;
      then (p = 1 or p = 2) & (q = 1 or q = 2) by A18,NUMBER05:21;
      hence thesis by A16,INT_2:def 4;
    end;
    suppose
A19:  4 divides n+3;
      consider p,q being Prime such that
A20:  p <> q and
A21:  n+3 = p*q by A6;
A22:  p divides p*q & q divides p*q;
      4 divides p*q by A21,A19;
      then p*q = 4 by GR_CY_3:1,XPRIMES0:4;
      then (p = 1 or p = 2) & (q = 1 or q = 2) by A22,NUMBER05:21;
      hence thesis by A20,INT_2:def 4;
    end;
  end;
