7703 = (2 * 3851) + 1 ;
hence not 2 divides 7703 by NAT_4:9; :: thesis:
7703 = (3 * 2567) + 2 ;
hence not 3 divides 7703 by NAT_4:9; :: thesis:
7703 = (5 * 1540) + 3 ;
hence not 5 divides 7703 by NAT_4:9; :: thesis:
7703 = (7 * 1100) + 3 ;
hence not 7 divides 7703 by NAT_4:9; :: thesis:
7703 = (11 * 700) + 3 ;
hence not 11 divides 7703 by NAT_4:9; :: thesis:
7703 = (13 * 592) + 7 ;
hence not 13 divides 7703 by NAT_4:9; :: thesis:
7703 = (17 * 453) + 2 ;
hence not 17 divides 7703 by NAT_4:9; :: thesis:
7703 = (19 * 405) + 8 ;
hence not 19 divides 7703 by NAT_4:9; :: thesis:
7703 = (23 * 334) + 21 ;
hence not 23 divides 7703 by NAT_4:9; :: thesis:
7703 = (29 * 265) + 18 ;
hence not 29 divides 7703 by NAT_4:9; :: thesis:
7703 = (31 * 248) + 15 ;
hence not 31 divides 7703 by NAT_4:9; :: thesis:
7703 = (37 * 208) + 7 ;
hence not 37 divides 7703 by NAT_4:9; :: thesis:
7703 = (41 * 187) + 36 ;
hence not 41 divides 7703 by NAT_4:9; :: thesis:
7703 = (43 * 179) + 6 ;
hence not 43 divides 7703 by NAT_4:9; :: thesis:
7703 = (47 * 163) + 42 ;
hence not 47 divides 7703 by NAT_4:9; :: thesis:
7703 = (53 * 145) + 18 ;
hence not 53 divides 7703 by NAT_4:9; :: thesis:
7703 = (59 * 130) + 33 ;
hence not 59 divides 7703 by NAT_4:9; :: thesis:
7703 = (61 * 126) + 17 ;
hence not 61 divides 7703 by NAT_4:9; :: thesis:
7703 = (67 * 114) + 65 ;
hence not 67 divides 7703 by NAT_4:9; :: thesis:
7703 = (71 * 108) + 35 ;
hence not 71 divides 7703 by NAT_4:9; :: thesis:
7703 = (73 * 105) + 38 ;
hence not 73 divides 7703 by NAT_4:9; :: thesis:
7703 = (79 * 97) + 40 ;
hence not 79 divides 7703 by NAT_4:9; :: thesis:
7703 = (83 * 92) + 67 ;
hence not 83 divides 7703 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7703 & n is prime holds
not n divides 7703 by XPRIMET1:46;
hence 7703 is prime by NAT_4:14; :: thesis: