7213 = (2 * 3606) + 1 ;
hence not 2 divides 7213 by NAT_4:9; :: thesis:
7213 = (3 * 2404) + 1 ;
hence not 3 divides 7213 by NAT_4:9; :: thesis:
7213 = (5 * 1442) + 3 ;
hence not 5 divides 7213 by NAT_4:9; :: thesis:
7213 = (7 * 1030) + 3 ;
hence not 7 divides 7213 by NAT_4:9; :: thesis:
7213 = (11 * 655) + 8 ;
hence not 11 divides 7213 by NAT_4:9; :: thesis:
7213 = (13 * 554) + 11 ;
hence not 13 divides 7213 by NAT_4:9; :: thesis:
7213 = (17 * 424) + 5 ;
hence not 17 divides 7213 by NAT_4:9; :: thesis:
7213 = (19 * 379) + 12 ;
hence not 19 divides 7213 by NAT_4:9; :: thesis:
7213 = (23 * 313) + 14 ;
hence not 23 divides 7213 by NAT_4:9; :: thesis:
7213 = (29 * 248) + 21 ;
hence not 29 divides 7213 by NAT_4:9; :: thesis:
7213 = (31 * 232) + 21 ;
hence not 31 divides 7213 by NAT_4:9; :: thesis:
7213 = (37 * 194) + 35 ;
hence not 37 divides 7213 by NAT_4:9; :: thesis:
7213 = (41 * 175) + 38 ;
hence not 41 divides 7213 by NAT_4:9; :: thesis:
7213 = (43 * 167) + 32 ;
hence not 43 divides 7213 by NAT_4:9; :: thesis:
7213 = (47 * 153) + 22 ;
hence not 47 divides 7213 by NAT_4:9; :: thesis:
7213 = (53 * 136) + 5 ;
hence not 53 divides 7213 by NAT_4:9; :: thesis:
7213 = (59 * 122) + 15 ;
hence not 59 divides 7213 by NAT_4:9; :: thesis:
7213 = (61 * 118) + 15 ;
hence not 61 divides 7213 by NAT_4:9; :: thesis:
7213 = (67 * 107) + 44 ;
hence not 67 divides 7213 by NAT_4:9; :: thesis:
7213 = (71 * 101) + 42 ;
hence not 71 divides 7213 by NAT_4:9; :: thesis:
7213 = (73 * 98) + 59 ;
hence not 73 divides 7213 by NAT_4:9; :: thesis:
7213 = (79 * 91) + 24 ;
hence not 79 divides 7213 by NAT_4:9; :: thesis:
7213 = (83 * 86) + 75 ;
hence not 83 divides 7213 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7213 & n is prime holds
not n divides 7213 by XPRIMET1:46;
hence 7213 is prime by NAT_4:14; :: thesis: