3527 = (2 * 1763) + 1 ;
hence not 2 divides 3527 by NAT_4:9; :: thesis:
3527 = (3 * 1175) + 2 ;
hence not 3 divides 3527 by NAT_4:9; :: thesis:
3527 = (5 * 705) + 2 ;
hence not 5 divides 3527 by NAT_4:9; :: thesis:
3527 = (7 * 503) + 6 ;
hence not 7 divides 3527 by NAT_4:9; :: thesis:
3527 = (11 * 320) + 7 ;
hence not 11 divides 3527 by NAT_4:9; :: thesis:
3527 = (13 * 271) + 4 ;
hence not 13 divides 3527 by NAT_4:9; :: thesis:
3527 = (17 * 207) + 8 ;
hence not 17 divides 3527 by NAT_4:9; :: thesis:
3527 = (19 * 185) + 12 ;
hence not 19 divides 3527 by NAT_4:9; :: thesis:
3527 = (23 * 153) + 8 ;
hence not 23 divides 3527 by NAT_4:9; :: thesis:
3527 = (29 * 121) + 18 ;
hence not 29 divides 3527 by NAT_4:9; :: thesis:
3527 = (31 * 113) + 24 ;
hence not 31 divides 3527 by NAT_4:9; :: thesis:
3527 = (37 * 95) + 12 ;
hence not 37 divides 3527 by NAT_4:9; :: thesis:
3527 = (41 * 86) + 1 ;
hence not 41 divides 3527 by NAT_4:9; :: thesis:
3527 = (43 * 82) + 1 ;
hence not 43 divides 3527 by NAT_4:9; :: thesis:
3527 = (47 * 75) + 2 ;
hence not 47 divides 3527 by NAT_4:9; :: thesis:
3527 = (53 * 66) + 29 ;
hence not 53 divides 3527 by NAT_4:9; :: thesis:
3527 = (59 * 59) + 46 ;
hence not 59 divides 3527 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3527 & n is prime holds
not n divides 3527 by XPRIMET1:34;
hence 3527 is prime by NAT_4:14; :: thesis: