reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve r for Real;
reserve c for Complex;
reserve e1,e2,e3,e4,e5 for ExtReal;

theorem
  {n where n is Nat: n+1 is prime & n+3 is prime & n+7 is prime &
   n+9 is prime & n+13 is prime & n+15 is prime} = {4}
  proof
    set A = {n where n is Nat: n+1 is prime & n+3 is prime & n+7 is prime &
    n+9 is prime & n+13 is prime & n+15 is prime};
A1: 5 = 4+1;
    thus A c= {4}
    proof
      let x be object;
      assume x in A;
      then consider n such that
A2:   x = n and
A3:   n+1 is prime and
A4:   n+3 is prime and
A5:   n+7 is prime and
A6:   n+9 is prime and
A7:   n+13 is prime and
A8:   n+15 is prime;
      now
        assume n <> 4;
        then per cases by XXREAL_0:1;
        suppose n < 3+1;
          then n <= 3 by NAT_1:13;
          then n = 0 or ... or n = 3;
          then per cases;
          suppose n = 0;
            hence contradiction by A3;
          end;
          suppose n = 1;
            hence contradiction by A4,INT_2:29;
          end;
          suppose n = 2;
            hence contradiction by A5,NAT_4:57;
          end;
          suppose n = 3;
            hence contradiction by A3,INT_2:29;
          end;
        end;
        suppose
A9:       n > 4;
          5 divides n+1 or 5 divides n+7 or 5 divides n+9 or 5 divides n+13 or
          5 divides n+15 by Th62;
          then 5 = n+1 or 5 = n+7 or 5 = n+9 or 5 = n+13 or 5 = n+15
          by A3,A5,A6,A7,A8;
          hence contradiction by A1,A9,XREAL_1:8;
        end;
      end;
      hence thesis by A2,TARSKI:def 1;
    end;
    let x be object;
    assume x in {4};
    then
A10: x = 4 by TARSKI:def 1;
    4+3 = 7 & 4+7 = 11 & 4+9 = 13 & 4+13 = 17 & 4+15 = 19;
    hence thesis by A1,A10,PEPIN:59,60,NAT_4:26,27,28,29;
  end;
