3739 = (2 * 1869) + 1 ;
hence not 2 divides 3739 by NAT_4:9; :: thesis:
3739 = (3 * 1246) + 1 ;
hence not 3 divides 3739 by NAT_4:9; :: thesis:
3739 = (5 * 747) + 4 ;
hence not 5 divides 3739 by NAT_4:9; :: thesis:
3739 = (7 * 534) + 1 ;
hence not 7 divides 3739 by NAT_4:9; :: thesis:
3739 = (11 * 339) + 10 ;
hence not 11 divides 3739 by NAT_4:9; :: thesis:
3739 = (13 * 287) + 8 ;
hence not 13 divides 3739 by NAT_4:9; :: thesis:
3739 = (17 * 219) + 16 ;
hence not 17 divides 3739 by NAT_4:9; :: thesis:
3739 = (19 * 196) + 15 ;
hence not 19 divides 3739 by NAT_4:9; :: thesis:
3739 = (23 * 162) + 13 ;
hence not 23 divides 3739 by NAT_4:9; :: thesis:
3739 = (29 * 128) + 27 ;
hence not 29 divides 3739 by NAT_4:9; :: thesis:
3739 = (31 * 120) + 19 ;
hence not 31 divides 3739 by NAT_4:9; :: thesis:
3739 = (37 * 101) + 2 ;
hence not 37 divides 3739 by NAT_4:9; :: thesis:
3739 = (41 * 91) + 8 ;
hence not 41 divides 3739 by NAT_4:9; :: thesis:
3739 = (43 * 86) + 41 ;
hence not 43 divides 3739 by NAT_4:9; :: thesis:
3739 = (47 * 79) + 26 ;
hence not 47 divides 3739 by NAT_4:9; :: thesis:
3739 = (53 * 70) + 29 ;
hence not 53 divides 3739 by NAT_4:9; :: thesis:
3739 = (59 * 63) + 22 ;
hence not 59 divides 3739 by NAT_4:9; :: thesis:
3739 = (61 * 61) + 18 ;
hence not 61 divides 3739 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3739 & n is prime holds
not n divides 3739 by XPRIMET1:36;
hence 3739 is prime by NAT_4:14; :: thesis: