1187 = (2 * 593) + 1 ;
hence not 2 divides 1187 by NAT_4:9; :: thesis:
1187 = (3 * 395) + 2 ;
hence not 3 divides 1187 by NAT_4:9; :: thesis:
1187 = (5 * 237) + 2 ;
hence not 5 divides 1187 by NAT_4:9; :: thesis:
1187 = (7 * 169) + 4 ;
hence not 7 divides 1187 by NAT_4:9; :: thesis:
1187 = (11 * 107) + 10 ;
hence not 11 divides 1187 by NAT_4:9; :: thesis:
1187 = (13 * 91) + 4 ;
hence not 13 divides 1187 by NAT_4:9; :: thesis:
1187 = (17 * 69) + 14 ;
hence not 17 divides 1187 by NAT_4:9; :: thesis:
1187 = (19 * 62) + 9 ;
hence not 19 divides 1187 by NAT_4:9; :: thesis:
1187 = (23 * 51) + 14 ;
hence not 23 divides 1187 by NAT_4:9; :: thesis:
1187 = (29 * 40) + 27 ;
hence not 29 divides 1187 by NAT_4:9; :: thesis:
1187 = (31 * 38) + 9 ;
hence not 31 divides 1187 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1187 & n is prime holds
not n divides 1187 by XPRIMET1:22;
hence 1187 is prime by NAT_4:14; :: thesis: