now :: thesis: ( not 2 divides 2663 & not 3 divides 2663 & not 5 divides 2663 & not 7 divides 2663 & not 11 divides 2663 & not 13 divides 2663 & not 17 divides 2663 & not 19 divides 2663 & not 23 divides 2663 & not 29 divides 2663 & not 31 divides 2663 & not 37 divides 2663 & not 41 divides 2663 & not 43 divides 2663 & not 47 divides 2663 )

2663 = (2 * 1331) + 1 ;
hence not 2 divides 2663 by NAT_4:9; :: thesis: ( not 3 divides 2663 & not 5 divides 2663 & not 7 divides 2663 & not 11 divides 2663 & not 13 divides 2663 & not 17 divides 2663 & not 19 divides 2663 & not 23 divides 2663 & not 29 divides 2663 & not 31 divides 2663 & not 37 divides 2663 & not 41 divides 2663 & not 43 divides 2663 & not 47 divides 2663 )
2663 = (3 * 887) + 2 ;
hence not 3 divides 2663 by NAT_4:9; :: thesis: ( not 5 divides 2663 & not 7 divides 2663 & not 11 divides 2663 & not 13 divides 2663 & not 17 divides 2663 & not 19 divides 2663 & not 23 divides 2663 & not 29 divides 2663 & not 31 divides 2663 & not 37 divides 2663 & not 41 divides 2663 & not 43 divides 2663 & not 47 divides 2663 )
2663 = (5 * 532) + 3 ;
hence not 5 divides 2663 by NAT_4:9; :: thesis: ( not 7 divides 2663 & not 11 divides 2663 & not 13 divides 2663 & not 17 divides 2663 & not 19 divides 2663 & not 23 divides 2663 & not 29 divides 2663 & not 31 divides 2663 & not 37 divides 2663 & not 41 divides 2663 & not 43 divides 2663 & not 47 divides 2663 )
2663 = (7 * 380) + 3 ;
hence not 7 divides 2663 by NAT_4:9; :: thesis: ( not 11 divides 2663 & not 13 divides 2663 & not 17 divides 2663 & not 19 divides 2663 & not 23 divides 2663 & not 29 divides 2663 & not 31 divides 2663 & not 37 divides 2663 & not 41 divides 2663 & not 43 divides 2663 & not 47 divides 2663 )
2663 = (11 * 242) + 1 ;
hence not 11 divides 2663 by NAT_4:9; :: thesis: ( not 13 divides 2663 & not 17 divides 2663 & not 19 divides 2663 & not 23 divides 2663 & not 29 divides 2663 & not 31 divides 2663 & not 37 divides 2663 & not 41 divides 2663 & not 43 divides 2663 & not 47 divides 2663 )
2663 = (13 * 204) + 11 ;
hence not 13 divides 2663 by NAT_4:9; :: thesis: ( not 17 divides 2663 & not 19 divides 2663 & not 23 divides 2663 & not 29 divides 2663 & not 31 divides 2663 & not 37 divides 2663 & not 41 divides 2663 & not 43 divides 2663 & not 47 divides 2663 )
2663 = (17 * 156) + 11 ;
hence not 17 divides 2663 by NAT_4:9; :: thesis: ( not 19 divides 2663 & not 23 divides 2663 & not 29 divides 2663 & not 31 divides 2663 & not 37 divides 2663 & not 41 divides 2663 & not 43 divides 2663 & not 47 divides 2663 )
2663 = (19 * 140) + 3 ;
hence not 19 divides 2663 by NAT_4:9; :: thesis: ( not 23 divides 2663 & not 29 divides 2663 & not 31 divides 2663 & not 37 divides 2663 & not 41 divides 2663 & not 43 divides 2663 & not 47 divides 2663 )
2663 = (23 * 115) + 18 ;
hence not 23 divides 2663 by NAT_4:9; :: thesis: ( not 29 divides 2663 & not 31 divides 2663 & not 37 divides 2663 & not 41 divides 2663 & not 43 divides 2663 & not 47 divides 2663 )
2663 = (29 * 91) + 24 ;
hence not 29 divides 2663 by NAT_4:9; :: thesis: ( not 31 divides 2663 & not 37 divides 2663 & not 41 divides 2663 & not 43 divides 2663 & not 47 divides 2663 )
2663 = (31 * 85) + 28 ;
hence not 31 divides 2663 by NAT_4:9; :: thesis: ( not 37 divides 2663 & not 41 divides 2663 & not 43 divides 2663 & not 47 divides 2663 )
2663 = (37 * 71) + 36 ;
hence not 37 divides 2663 by NAT_4:9; :: thesis: ( not 41 divides 2663 & not 43 divides 2663 & not 47 divides 2663 )
2663 = (41 * 64) + 39 ;
hence not 41 divides 2663 by NAT_4:9; :: thesis: ( not 43 divides 2663 & not 47 divides 2663 )
2663 = (43 * 61) + 40 ;
hence not 43 divides 2663 by NAT_4:9; :: thesis: not 47 divides 2663
2663 = (47 * 56) + 31 ;
hence not 47 divides 2663 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 2663 & n is prime holds
not n divides 2663 by XPRIMET1:30;
hence 2663 is prime by NAT_4:14; :: thesis: verum