
theorem
  5189 is prime
proof
  now
    5189 = 2*2594 + 1; hence not 2 divides 5189 by NAT_4:9;
    5189 = 3*1729 + 2; hence not 3 divides 5189 by NAT_4:9;
    5189 = 5*1037 + 4; hence not 5 divides 5189 by NAT_4:9;
    5189 = 7*741 + 2; hence not 7 divides 5189 by NAT_4:9;
    5189 = 11*471 + 8; hence not 11 divides 5189 by NAT_4:9;
    5189 = 13*399 + 2; hence not 13 divides 5189 by NAT_4:9;
    5189 = 17*305 + 4; hence not 17 divides 5189 by NAT_4:9;
    5189 = 19*273 + 2; hence not 19 divides 5189 by NAT_4:9;
    5189 = 23*225 + 14; hence not 23 divides 5189 by NAT_4:9;
    5189 = 29*178 + 27; hence not 29 divides 5189 by NAT_4:9;
    5189 = 31*167 + 12; hence not 31 divides 5189 by NAT_4:9;
    5189 = 37*140 + 9; hence not 37 divides 5189 by NAT_4:9;
    5189 = 41*126 + 23; hence not 41 divides 5189 by NAT_4:9;
    5189 = 43*120 + 29; hence not 43 divides 5189 by NAT_4:9;
    5189 = 47*110 + 19; hence not 47 divides 5189 by NAT_4:9;
    5189 = 53*97 + 48; hence not 53 divides 5189 by NAT_4:9;
    5189 = 59*87 + 56; hence not 59 divides 5189 by NAT_4:9;
    5189 = 61*85 + 4; hence not 61 divides 5189 by NAT_4:9;
    5189 = 67*77 + 30; hence not 67 divides 5189 by NAT_4:9;
    5189 = 71*73 + 6; hence not 71 divides 5189 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 5189 & n is prime
  holds not n divides 5189 by XPRIMET1:40;
  hence thesis by NAT_4:14;
end;
