9623 = (2 * 4811) + 1 ;
hence not 2 divides 9623 by NAT_4:9; :: thesis:
9623 = (3 * 3207) + 2 ;
hence not 3 divides 9623 by NAT_4:9; :: thesis:
9623 = (5 * 1924) + 3 ;
hence not 5 divides 9623 by NAT_4:9; :: thesis:
9623 = (7 * 1374) + 5 ;
hence not 7 divides 9623 by NAT_4:9; :: thesis:
9623 = (11 * 874) + 9 ;
hence not 11 divides 9623 by NAT_4:9; :: thesis:
9623 = (13 * 740) + 3 ;
hence not 13 divides 9623 by NAT_4:9; :: thesis:
9623 = (17 * 566) + 1 ;
hence not 17 divides 9623 by NAT_4:9; :: thesis:
9623 = (19 * 506) + 9 ;
hence not 19 divides 9623 by NAT_4:9; :: thesis:
9623 = (23 * 418) + 9 ;
hence not 23 divides 9623 by NAT_4:9; :: thesis:
9623 = (29 * 331) + 24 ;
hence not 29 divides 9623 by NAT_4:9; :: thesis:
9623 = (31 * 310) + 13 ;
hence not 31 divides 9623 by NAT_4:9; :: thesis:
9623 = (37 * 260) + 3 ;
hence not 37 divides 9623 by NAT_4:9; :: thesis:
9623 = (41 * 234) + 29 ;
hence not 41 divides 9623 by NAT_4:9; :: thesis:
9623 = (43 * 223) + 34 ;
hence not 43 divides 9623 by NAT_4:9; :: thesis:
9623 = (47 * 204) + 35 ;
hence not 47 divides 9623 by NAT_4:9; :: thesis:
9623 = (53 * 181) + 30 ;
hence not 53 divides 9623 by NAT_4:9; :: thesis:
9623 = (59 * 163) + 6 ;
hence not 59 divides 9623 by NAT_4:9; :: thesis:
9623 = (61 * 157) + 46 ;
hence not 61 divides 9623 by NAT_4:9; :: thesis:
9623 = (67 * 143) + 42 ;
hence not 67 divides 9623 by NAT_4:9; :: thesis:
9623 = (71 * 135) + 38 ;
hence not 71 divides 9623 by NAT_4:9; :: thesis:
9623 = (73 * 131) + 60 ;
hence not 73 divides 9623 by NAT_4:9; :: thesis:
9623 = (79 * 121) + 64 ;
hence not 79 divides 9623 by NAT_4:9; :: thesis:
9623 = (83 * 115) + 78 ;
hence not 83 divides 9623 by NAT_4:9; :: thesis:
9623 = (89 * 108) + 11 ;
hence not 89 divides 9623 by NAT_4:9; :: thesis:
9623 = (97 * 99) + 20 ;
hence not 97 divides 9623 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 9623 & n is prime holds
not n divides 9623 by XPRIMET1:50;
hence 9623 is prime by NAT_4:14; :: thesis: