1783 = (2 * 891) + 1 ;
hence not 2 divides 1783 by NAT_4:9; :: thesis:
1783 = (3 * 594) + 1 ;
hence not 3 divides 1783 by NAT_4:9; :: thesis:
1783 = (5 * 356) + 3 ;
hence not 5 divides 1783 by NAT_4:9; :: thesis:
1783 = (7 * 254) + 5 ;
hence not 7 divides 1783 by NAT_4:9; :: thesis:
1783 = (11 * 162) + 1 ;
hence not 11 divides 1783 by NAT_4:9; :: thesis:
1783 = (13 * 137) + 2 ;
hence not 13 divides 1783 by NAT_4:9; :: thesis:
1783 = (17 * 104) + 15 ;
hence not 17 divides 1783 by NAT_4:9; :: thesis:
1783 = (19 * 93) + 16 ;
hence not 19 divides 1783 by NAT_4:9; :: thesis:
1783 = (23 * 77) + 12 ;
hence not 23 divides 1783 by NAT_4:9; :: thesis:
1783 = (29 * 61) + 14 ;
hence not 29 divides 1783 by NAT_4:9; :: thesis:
1783 = (31 * 57) + 16 ;
hence not 31 divides 1783 by NAT_4:9; :: thesis:
1783 = (37 * 48) + 7 ;
hence not 37 divides 1783 by NAT_4:9; :: thesis:
1783 = (41 * 43) + 20 ;
hence not 41 divides 1783 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1783 & n is prime holds
not n divides 1783 by XPRIMET1:26;
hence 1783 is prime by NAT_4:14; :: thesis: