4723 = (2 * 2361) + 1 ;
hence not 2 divides 4723 by NAT_4:9; :: thesis:
4723 = (3 * 1574) + 1 ;
hence not 3 divides 4723 by NAT_4:9; :: thesis:
4723 = (5 * 944) + 3 ;
hence not 5 divides 4723 by NAT_4:9; :: thesis:
4723 = (7 * 674) + 5 ;
hence not 7 divides 4723 by NAT_4:9; :: thesis:
4723 = (11 * 429) + 4 ;
hence not 11 divides 4723 by NAT_4:9; :: thesis:
4723 = (13 * 363) + 4 ;
hence not 13 divides 4723 by NAT_4:9; :: thesis:
4723 = (17 * 277) + 14 ;
hence not 17 divides 4723 by NAT_4:9; :: thesis:
4723 = (19 * 248) + 11 ;
hence not 19 divides 4723 by NAT_4:9; :: thesis:
4723 = (23 * 205) + 8 ;
hence not 23 divides 4723 by NAT_4:9; :: thesis:
4723 = (29 * 162) + 25 ;
hence not 29 divides 4723 by NAT_4:9; :: thesis:
4723 = (31 * 152) + 11 ;
hence not 31 divides 4723 by NAT_4:9; :: thesis:
4723 = (37 * 127) + 24 ;
hence not 37 divides 4723 by NAT_4:9; :: thesis:
4723 = (41 * 115) + 8 ;
hence not 41 divides 4723 by NAT_4:9; :: thesis:
4723 = (43 * 109) + 36 ;
hence not 43 divides 4723 by NAT_4:9; :: thesis:
4723 = (47 * 100) + 23 ;
hence not 47 divides 4723 by NAT_4:9; :: thesis:
4723 = (53 * 89) + 6 ;
hence not 53 divides 4723 by NAT_4:9; :: thesis:
4723 = (59 * 80) + 3 ;
hence not 59 divides 4723 by NAT_4:9; :: thesis:
4723 = (61 * 77) + 26 ;
hence not 61 divides 4723 by NAT_4:9; :: thesis:
4723 = (67 * 70) + 33 ;
hence not 67 divides 4723 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4723 & n is prime holds
not n divides 4723 by XPRIMET1:38;
hence 4723 is prime by NAT_4:14; :: thesis: