5623 = (2 * 2811) + 1 ;
hence not 2 divides 5623 by NAT_4:9; :: thesis:
5623 = (3 * 1874) + 1 ;
hence not 3 divides 5623 by NAT_4:9; :: thesis:
5623 = (5 * 1124) + 3 ;
hence not 5 divides 5623 by NAT_4:9; :: thesis:
5623 = (7 * 803) + 2 ;
hence not 7 divides 5623 by NAT_4:9; :: thesis:
5623 = (11 * 511) + 2 ;
hence not 11 divides 5623 by NAT_4:9; :: thesis:
5623 = (13 * 432) + 7 ;
hence not 13 divides 5623 by NAT_4:9; :: thesis:
5623 = (17 * 330) + 13 ;
hence not 17 divides 5623 by NAT_4:9; :: thesis:
5623 = (19 * 295) + 18 ;
hence not 19 divides 5623 by NAT_4:9; :: thesis:
5623 = (23 * 244) + 11 ;
hence not 23 divides 5623 by NAT_4:9; :: thesis:
5623 = (29 * 193) + 26 ;
hence not 29 divides 5623 by NAT_4:9; :: thesis:
5623 = (31 * 181) + 12 ;
hence not 31 divides 5623 by NAT_4:9; :: thesis:
5623 = (37 * 151) + 36 ;
hence not 37 divides 5623 by NAT_4:9; :: thesis:
5623 = (41 * 137) + 6 ;
hence not 41 divides 5623 by NAT_4:9; :: thesis:
5623 = (43 * 130) + 33 ;
hence not 43 divides 5623 by NAT_4:9; :: thesis:
5623 = (47 * 119) + 30 ;
hence not 47 divides 5623 by NAT_4:9; :: thesis:
5623 = (53 * 106) + 5 ;
hence not 53 divides 5623 by NAT_4:9; :: thesis:
5623 = (59 * 95) + 18 ;
hence not 59 divides 5623 by NAT_4:9; :: thesis:
5623 = (61 * 92) + 11 ;
hence not 61 divides 5623 by NAT_4:9; :: thesis:
5623 = (67 * 83) + 62 ;
hence not 67 divides 5623 by NAT_4:9; :: thesis:
5623 = (71 * 79) + 14 ;
hence not 71 divides 5623 by NAT_4:9; :: thesis:
5623 = (73 * 77) + 2 ;
hence not 73 divides 5623 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5623 & n is prime holds
not n divides 5623 by XPRIMET1:42;
hence 5623 is prime by NAT_4:14; :: thesis: