5827 = (2 * 2913) + 1 ;
hence not 2 divides 5827 by NAT_4:9; :: thesis:
5827 = (3 * 1942) + 1 ;
hence not 3 divides 5827 by NAT_4:9; :: thesis:
5827 = (5 * 1165) + 2 ;
hence not 5 divides 5827 by NAT_4:9; :: thesis:
5827 = (7 * 832) + 3 ;
hence not 7 divides 5827 by NAT_4:9; :: thesis:
5827 = (11 * 529) + 8 ;
hence not 11 divides 5827 by NAT_4:9; :: thesis:
5827 = (13 * 448) + 3 ;
hence not 13 divides 5827 by NAT_4:9; :: thesis:
5827 = (17 * 342) + 13 ;
hence not 17 divides 5827 by NAT_4:9; :: thesis:
5827 = (19 * 306) + 13 ;
hence not 19 divides 5827 by NAT_4:9; :: thesis:
5827 = (23 * 253) + 8 ;
hence not 23 divides 5827 by NAT_4:9; :: thesis:
5827 = (29 * 200) + 27 ;
hence not 29 divides 5827 by NAT_4:9; :: thesis:
5827 = (31 * 187) + 30 ;
hence not 31 divides 5827 by NAT_4:9; :: thesis:
5827 = (37 * 157) + 18 ;
hence not 37 divides 5827 by NAT_4:9; :: thesis:
5827 = (41 * 142) + 5 ;
hence not 41 divides 5827 by NAT_4:9; :: thesis:
5827 = (43 * 135) + 22 ;
hence not 43 divides 5827 by NAT_4:9; :: thesis:
5827 = (47 * 123) + 46 ;
hence not 47 divides 5827 by NAT_4:9; :: thesis:
5827 = (53 * 109) + 50 ;
hence not 53 divides 5827 by NAT_4:9; :: thesis:
5827 = (59 * 98) + 45 ;
hence not 59 divides 5827 by NAT_4:9; :: thesis:
5827 = (61 * 95) + 32 ;
hence not 61 divides 5827 by NAT_4:9; :: thesis:
5827 = (67 * 86) + 65 ;
hence not 67 divides 5827 by NAT_4:9; :: thesis:
5827 = (71 * 82) + 5 ;
hence not 71 divides 5827 by NAT_4:9; :: thesis:
5827 = (73 * 79) + 60 ;
hence not 73 divides 5827 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5827 & n is prime holds
not n divides 5827 by XPRIMET1:42;
hence 5827 is prime by NAT_4:14; :: thesis: