4547 = (2 * 2273) + 1 ;
hence not 2 divides 4547 by NAT_4:9; :: thesis:
4547 = (3 * 1515) + 2 ;
hence not 3 divides 4547 by NAT_4:9; :: thesis:
4547 = (5 * 909) + 2 ;
hence not 5 divides 4547 by NAT_4:9; :: thesis:
4547 = (7 * 649) + 4 ;
hence not 7 divides 4547 by NAT_4:9; :: thesis:
4547 = (11 * 413) + 4 ;
hence not 11 divides 4547 by NAT_4:9; :: thesis:
4547 = (13 * 349) + 10 ;
hence not 13 divides 4547 by NAT_4:9; :: thesis:
4547 = (17 * 267) + 8 ;
hence not 17 divides 4547 by NAT_4:9; :: thesis:
4547 = (19 * 239) + 6 ;
hence not 19 divides 4547 by NAT_4:9; :: thesis:
4547 = (23 * 197) + 16 ;
hence not 23 divides 4547 by NAT_4:9; :: thesis:
4547 = (29 * 156) + 23 ;
hence not 29 divides 4547 by NAT_4:9; :: thesis:
4547 = (31 * 146) + 21 ;
hence not 31 divides 4547 by NAT_4:9; :: thesis:
4547 = (37 * 122) + 33 ;
hence not 37 divides 4547 by NAT_4:9; :: thesis:
4547 = (41 * 110) + 37 ;
hence not 41 divides 4547 by NAT_4:9; :: thesis:
4547 = (43 * 105) + 32 ;
hence not 43 divides 4547 by NAT_4:9; :: thesis:
4547 = (47 * 96) + 35 ;
hence not 47 divides 4547 by NAT_4:9; :: thesis:
4547 = (53 * 85) + 42 ;
hence not 53 divides 4547 by NAT_4:9; :: thesis:
4547 = (59 * 77) + 4 ;
hence not 59 divides 4547 by NAT_4:9; :: thesis:
4547 = (61 * 74) + 33 ;
hence not 61 divides 4547 by NAT_4:9; :: thesis:
4547 = (67 * 67) + 58 ;
hence not 67 divides 4547 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4547 & n is prime holds
not n divides 4547 by XPRIMET1:38;
hence 4547 is prime by NAT_4:14; :: thesis: