7757 = (2 * 3878) + 1 ;
hence not 2 divides 7757 by NAT_4:9; :: thesis:
7757 = (3 * 2585) + 2 ;
hence not 3 divides 7757 by NAT_4:9; :: thesis:
7757 = (5 * 1551) + 2 ;
hence not 5 divides 7757 by NAT_4:9; :: thesis:
7757 = (7 * 1108) + 1 ;
hence not 7 divides 7757 by NAT_4:9; :: thesis:
7757 = (11 * 705) + 2 ;
hence not 11 divides 7757 by NAT_4:9; :: thesis:
7757 = (13 * 596) + 9 ;
hence not 13 divides 7757 by NAT_4:9; :: thesis:
7757 = (17 * 456) + 5 ;
hence not 17 divides 7757 by NAT_4:9; :: thesis:
7757 = (19 * 408) + 5 ;
hence not 19 divides 7757 by NAT_4:9; :: thesis:
7757 = (23 * 337) + 6 ;
hence not 23 divides 7757 by NAT_4:9; :: thesis:
7757 = (29 * 267) + 14 ;
hence not 29 divides 7757 by NAT_4:9; :: thesis:
7757 = (31 * 250) + 7 ;
hence not 31 divides 7757 by NAT_4:9; :: thesis:
7757 = (37 * 209) + 24 ;
hence not 37 divides 7757 by NAT_4:9; :: thesis:
7757 = (41 * 189) + 8 ;
hence not 41 divides 7757 by NAT_4:9; :: thesis:
7757 = (43 * 180) + 17 ;
hence not 43 divides 7757 by NAT_4:9; :: thesis:
7757 = (47 * 165) + 2 ;
hence not 47 divides 7757 by NAT_4:9; :: thesis:
7757 = (53 * 146) + 19 ;
hence not 53 divides 7757 by NAT_4:9; :: thesis:
7757 = (59 * 131) + 28 ;
hence not 59 divides 7757 by NAT_4:9; :: thesis:
7757 = (61 * 127) + 10 ;
hence not 61 divides 7757 by NAT_4:9; :: thesis:
7757 = (67 * 115) + 52 ;
hence not 67 divides 7757 by NAT_4:9; :: thesis:
7757 = (71 * 109) + 18 ;
hence not 71 divides 7757 by NAT_4:9; :: thesis:
7757 = (73 * 106) + 19 ;
hence not 73 divides 7757 by NAT_4:9; :: thesis:
7757 = (79 * 98) + 15 ;
hence not 79 divides 7757 by NAT_4:9; :: thesis:
7757 = (83 * 93) + 38 ;
hence not 83 divides 7757 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7757 & n is prime holds
not n divides 7757 by XPRIMET1:46;
hence 7757 is prime by NAT_4:14; :: thesis: