
theorem Lemma73:
  for k being Nat st k >= 1 holds
    30 * k + 7 is not_representable_by_sum_or_difference_of_two_primes
  proof
    let k be Nat;
    assume
AA: k >= 1;
    assume
    not 30 * k + 7 is not_representable_by_sum_or_difference_of_two_primes;
      then
    consider p,q being Prime such that
A1: 30 * k + 7 = p + q or 30 * k + 7 = p - q;
    p is even or q is even by A1,Lemma30kOdd; then
    per cases by INT_2:def 4,ABIAN:def 1;
    suppose
S1:   q = 2;
      per cases by A1;
      suppose
ss:     30 * k + 7 = p + q; then
        p = 5 * (6 * k + 1) by S1; then
        5 divides p; then
        p = 5 by INT_2:def 4; then
        k = 0 by S1,ss;
        hence thesis by AA;
      end;
      suppose
ss:     30 * k + 7 = p - q; then
        p = 3 * (10 * k + 3) by S1; then
        3 divides p; then
s0:     p = 3 by INT_2:def 4;
        10 * k >= 10 * 1 by AA,XREAL_1:66; then
        10 * k + 3 >= 10 * 1 + 3 by XREAL_1:6;
        hence thesis by s0,ss,S1;
      end;
    end;
    suppose
S1:   p = 2;
      per cases by A1;
      suppose
s0:     30 * k + 7 = p + q; then
        q = 5 * (6 * k + 1) by S1; then
        5 divides q; then
        q = 5 by INT_2:def 4; then
        k = 0 by s0,S1;
        hence thesis by AA;
      end;
      suppose
        30 * k + 7 = p - q; then
        -q = 30 * k + 5 by S1;
        hence thesis;
      end;
    end;
  end;
