2053 = (2 * 1026) + 1 ;
hence not 2 divides 2053 by NAT_4:9; :: thesis:
2053 = (3 * 684) + 1 ;
hence not 3 divides 2053 by NAT_4:9; :: thesis:
2053 = (5 * 410) + 3 ;
hence not 5 divides 2053 by NAT_4:9; :: thesis:
2053 = (7 * 293) + 2 ;
hence not 7 divides 2053 by NAT_4:9; :: thesis:
2053 = (11 * 186) + 7 ;
hence not 11 divides 2053 by NAT_4:9; :: thesis:
2053 = (13 * 157) + 12 ;
hence not 13 divides 2053 by NAT_4:9; :: thesis:
2053 = (17 * 120) + 13 ;
hence not 17 divides 2053 by NAT_4:9; :: thesis:
2053 = (19 * 108) + 1 ;
hence not 19 divides 2053 by NAT_4:9; :: thesis:
2053 = (23 * 89) + 6 ;
hence not 23 divides 2053 by NAT_4:9; :: thesis:
2053 = (29 * 70) + 23 ;
hence not 29 divides 2053 by NAT_4:9; :: thesis:
2053 = (31 * 66) + 7 ;
hence not 31 divides 2053 by NAT_4:9; :: thesis:
2053 = (37 * 55) + 18 ;
hence not 37 divides 2053 by NAT_4:9; :: thesis:
2053 = (41 * 50) + 3 ;
hence not 41 divides 2053 by NAT_4:9; :: thesis:
2053 = (43 * 47) + 32 ;
hence not 43 divides 2053 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2053 & n is prime holds
not n divides 2053 by XPRIMET1:28;
hence 2053 is prime by NAT_4:14; :: thesis: