1753 = (2 * 876) + 1 ;
hence not 2 divides 1753 by NAT_4:9; :: thesis:
1753 = (3 * 584) + 1 ;
hence not 3 divides 1753 by NAT_4:9; :: thesis:
1753 = (5 * 350) + 3 ;
hence not 5 divides 1753 by NAT_4:9; :: thesis:
1753 = (7 * 250) + 3 ;
hence not 7 divides 1753 by NAT_4:9; :: thesis:
1753 = (11 * 159) + 4 ;
hence not 11 divides 1753 by NAT_4:9; :: thesis:
1753 = (13 * 134) + 11 ;
hence not 13 divides 1753 by NAT_4:9; :: thesis:
1753 = (17 * 103) + 2 ;
hence not 17 divides 1753 by NAT_4:9; :: thesis:
1753 = (19 * 92) + 5 ;
hence not 19 divides 1753 by NAT_4:9; :: thesis:
1753 = (23 * 76) + 5 ;
hence not 23 divides 1753 by NAT_4:9; :: thesis:
1753 = (29 * 60) + 13 ;
hence not 29 divides 1753 by NAT_4:9; :: thesis:
1753 = (31 * 56) + 17 ;
hence not 31 divides 1753 by NAT_4:9; :: thesis:
1753 = (37 * 47) + 14 ;
hence not 37 divides 1753 by NAT_4:9; :: thesis:
1753 = (41 * 42) + 31 ;
hence not 41 divides 1753 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1753 & n is prime holds
not n divides 1753 by XPRIMET1:26;
hence 1753 is prime by NAT_4:14; :: thesis: