2741 = (2 * 1370) + 1 ;
hence not 2 divides 2741 by NAT_4:9; :: thesis:
2741 = (3 * 913) + 2 ;
hence not 3 divides 2741 by NAT_4:9; :: thesis:
2741 = (5 * 548) + 1 ;
hence not 5 divides 2741 by NAT_4:9; :: thesis:
2741 = (7 * 391) + 4 ;
hence not 7 divides 2741 by NAT_4:9; :: thesis:
2741 = (11 * 249) + 2 ;
hence not 11 divides 2741 by NAT_4:9; :: thesis:
2741 = (13 * 210) + 11 ;
hence not 13 divides 2741 by NAT_4:9; :: thesis:
2741 = (17 * 161) + 4 ;
hence not 17 divides 2741 by NAT_4:9; :: thesis:
2741 = (19 * 144) + 5 ;
hence not 19 divides 2741 by NAT_4:9; :: thesis:
2741 = (23 * 119) + 4 ;
hence not 23 divides 2741 by NAT_4:9; :: thesis:
2741 = (29 * 94) + 15 ;
hence not 29 divides 2741 by NAT_4:9; :: thesis:
2741 = (31 * 88) + 13 ;
hence not 31 divides 2741 by NAT_4:9; :: thesis:
2741 = (37 * 74) + 3 ;
hence not 37 divides 2741 by NAT_4:9; :: thesis:
2741 = (41 * 66) + 35 ;
hence not 41 divides 2741 by NAT_4:9; :: thesis:
2741 = (43 * 63) + 32 ;
hence not 43 divides 2741 by NAT_4:9; :: thesis:
2741 = (47 * 58) + 15 ;
hence not 47 divides 2741 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2741 & n is prime holds
not n divides 2741 by XPRIMET1:30;
hence 2741 is prime by NAT_4:14; :: thesis: