5051 = (2 * 2525) + 1 ;
hence not 2 divides 5051 by NAT_4:9; :: thesis:
5051 = (3 * 1683) + 2 ;
hence not 3 divides 5051 by NAT_4:9; :: thesis:
5051 = (5 * 1010) + 1 ;
hence not 5 divides 5051 by NAT_4:9; :: thesis:
5051 = (7 * 721) + 4 ;
hence not 7 divides 5051 by NAT_4:9; :: thesis:
5051 = (11 * 459) + 2 ;
hence not 11 divides 5051 by NAT_4:9; :: thesis:
5051 = (13 * 388) + 7 ;
hence not 13 divides 5051 by NAT_4:9; :: thesis:
5051 = (17 * 297) + 2 ;
hence not 17 divides 5051 by NAT_4:9; :: thesis:
5051 = (19 * 265) + 16 ;
hence not 19 divides 5051 by NAT_4:9; :: thesis:
5051 = (23 * 219) + 14 ;
hence not 23 divides 5051 by NAT_4:9; :: thesis:
5051 = (29 * 174) + 5 ;
hence not 29 divides 5051 by NAT_4:9; :: thesis:
5051 = (31 * 162) + 29 ;
hence not 31 divides 5051 by NAT_4:9; :: thesis:
5051 = (37 * 136) + 19 ;
hence not 37 divides 5051 by NAT_4:9; :: thesis:
5051 = (41 * 123) + 8 ;
hence not 41 divides 5051 by NAT_4:9; :: thesis:
5051 = (43 * 117) + 20 ;
hence not 43 divides 5051 by NAT_4:9; :: thesis:
5051 = (47 * 107) + 22 ;
hence not 47 divides 5051 by NAT_4:9; :: thesis:
5051 = (53 * 95) + 16 ;
hence not 53 divides 5051 by NAT_4:9; :: thesis:
5051 = (59 * 85) + 36 ;
hence not 59 divides 5051 by NAT_4:9; :: thesis:
5051 = (61 * 82) + 49 ;
hence not 61 divides 5051 by NAT_4:9; :: thesis:
5051 = (67 * 75) + 26 ;
hence not 67 divides 5051 by NAT_4:9; :: thesis:
5051 = (71 * 71) + 10 ;
hence not 71 divides 5051 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5051 & n is prime holds
not n divides 5051 by XPRIMET1:40;
hence 5051 is prime by NAT_4:14; :: thesis: