now :: thesis: ( not 2 divides 4967 & not 3 divides 4967 & not 5 divides 4967 & not 7 divides 4967 & not 11 divides 4967 & not 13 divides 4967 & not 17 divides 4967 & not 19 divides 4967 & not 23 divides 4967 & not 29 divides 4967 & not 31 divides 4967 & not 37 divides 4967 & not 41 divides 4967 & not 43 divides 4967 & not 47 divides 4967 & not 53 divides 4967 & not 59 divides 4967 & not 61 divides 4967 & not 67 divides 4967 )

4967 = (2 * 2483) + 1 ;
hence not 2 divides 4967 by NAT_4:9; :: thesis: ( not 3 divides 4967 & not 5 divides 4967 & not 7 divides 4967 & not 11 divides 4967 & not 13 divides 4967 & not 17 divides 4967 & not 19 divides 4967 & not 23 divides 4967 & not 29 divides 4967 & not 31 divides 4967 & not 37 divides 4967 & not 41 divides 4967 & not 43 divides 4967 & not 47 divides 4967 & not 53 divides 4967 & not 59 divides 4967 & not 61 divides 4967 & not 67 divides 4967 )
4967 = (3 * 1655) + 2 ;
hence not 3 divides 4967 by NAT_4:9; :: thesis: ( not 5 divides 4967 & not 7 divides 4967 & not 11 divides 4967 & not 13 divides 4967 & not 17 divides 4967 & not 19 divides 4967 & not 23 divides 4967 & not 29 divides 4967 & not 31 divides 4967 & not 37 divides 4967 & not 41 divides 4967 & not 43 divides 4967 & not 47 divides 4967 & not 53 divides 4967 & not 59 divides 4967 & not 61 divides 4967 & not 67 divides 4967 )
4967 = (5 * 993) + 2 ;
hence not 5 divides 4967 by NAT_4:9; :: thesis: ( not 7 divides 4967 & not 11 divides 4967 & not 13 divides 4967 & not 17 divides 4967 & not 19 divides 4967 & not 23 divides 4967 & not 29 divides 4967 & not 31 divides 4967 & not 37 divides 4967 & not 41 divides 4967 & not 43 divides 4967 & not 47 divides 4967 & not 53 divides 4967 & not 59 divides 4967 & not 61 divides 4967 & not 67 divides 4967 )
4967 = (7 * 709) + 4 ;
hence not 7 divides 4967 by NAT_4:9; :: thesis: ( not 11 divides 4967 & not 13 divides 4967 & not 17 divides 4967 & not 19 divides 4967 & not 23 divides 4967 & not 29 divides 4967 & not 31 divides 4967 & not 37 divides 4967 & not 41 divides 4967 & not 43 divides 4967 & not 47 divides 4967 & not 53 divides 4967 & not 59 divides 4967 & not 61 divides 4967 & not 67 divides 4967 )
4967 = (11 * 451) + 6 ;
hence not 11 divides 4967 by NAT_4:9; :: thesis: ( not 13 divides 4967 & not 17 divides 4967 & not 19 divides 4967 & not 23 divides 4967 & not 29 divides 4967 & not 31 divides 4967 & not 37 divides 4967 & not 41 divides 4967 & not 43 divides 4967 & not 47 divides 4967 & not 53 divides 4967 & not 59 divides 4967 & not 61 divides 4967 & not 67 divides 4967 )
4967 = (13 * 382) + 1 ;
hence not 13 divides 4967 by NAT_4:9; :: thesis: ( not 17 divides 4967 & not 19 divides 4967 & not 23 divides 4967 & not 29 divides 4967 & not 31 divides 4967 & not 37 divides 4967 & not 41 divides 4967 & not 43 divides 4967 & not 47 divides 4967 & not 53 divides 4967 & not 59 divides 4967 & not 61 divides 4967 & not 67 divides 4967 )
4967 = (17 * 292) + 3 ;
hence not 17 divides 4967 by NAT_4:9; :: thesis: ( not 19 divides 4967 & not 23 divides 4967 & not 29 divides 4967 & not 31 divides 4967 & not 37 divides 4967 & not 41 divides 4967 & not 43 divides 4967 & not 47 divides 4967 & not 53 divides 4967 & not 59 divides 4967 & not 61 divides 4967 & not 67 divides 4967 )
4967 = (19 * 261) + 8 ;
hence not 19 divides 4967 by NAT_4:9; :: thesis: ( not 23 divides 4967 & not 29 divides 4967 & not 31 divides 4967 & not 37 divides 4967 & not 41 divides 4967 & not 43 divides 4967 & not 47 divides 4967 & not 53 divides 4967 & not 59 divides 4967 & not 61 divides 4967 & not 67 divides 4967 )
4967 = (23 * 215) + 22 ;
hence not 23 divides 4967 by NAT_4:9; :: thesis: ( not 29 divides 4967 & not 31 divides 4967 & not 37 divides 4967 & not 41 divides 4967 & not 43 divides 4967 & not 47 divides 4967 & not 53 divides 4967 & not 59 divides 4967 & not 61 divides 4967 & not 67 divides 4967 )
4967 = (29 * 171) + 8 ;
hence not 29 divides 4967 by NAT_4:9; :: thesis: ( not 31 divides 4967 & not 37 divides 4967 & not 41 divides 4967 & not 43 divides 4967 & not 47 divides 4967 & not 53 divides 4967 & not 59 divides 4967 & not 61 divides 4967 & not 67 divides 4967 )
4967 = (31 * 160) + 7 ;
hence not 31 divides 4967 by NAT_4:9; :: thesis: ( not 37 divides 4967 & not 41 divides 4967 & not 43 divides 4967 & not 47 divides 4967 & not 53 divides 4967 & not 59 divides 4967 & not 61 divides 4967 & not 67 divides 4967 )
4967 = (37 * 134) + 9 ;
hence not 37 divides 4967 by NAT_4:9; :: thesis: ( not 41 divides 4967 & not 43 divides 4967 & not 47 divides 4967 & not 53 divides 4967 & not 59 divides 4967 & not 61 divides 4967 & not 67 divides 4967 )
4967 = (41 * 121) + 6 ;
hence not 41 divides 4967 by NAT_4:9; :: thesis: ( not 43 divides 4967 & not 47 divides 4967 & not 53 divides 4967 & not 59 divides 4967 & not 61 divides 4967 & not 67 divides 4967 )
4967 = (43 * 115) + 22 ;
hence not 43 divides 4967 by NAT_4:9; :: thesis: ( not 47 divides 4967 & not 53 divides 4967 & not 59 divides 4967 & not 61 divides 4967 & not 67 divides 4967 )
4967 = (47 * 105) + 32 ;
hence not 47 divides 4967 by NAT_4:9; :: thesis: ( not 53 divides 4967 & not 59 divides 4967 & not 61 divides 4967 & not 67 divides 4967 )
4967 = (53 * 93) + 38 ;
hence not 53 divides 4967 by NAT_4:9; :: thesis: ( not 59 divides 4967 & not 61 divides 4967 & not 67 divides 4967 )
4967 = (59 * 84) + 11 ;
hence not 59 divides 4967 by NAT_4:9; :: thesis: ( not 61 divides 4967 & not 67 divides 4967 )
4967 = (61 * 81) + 26 ;
hence not 61 divides 4967 by NAT_4:9; :: thesis: not 67 divides 4967
4967 = (67 * 74) + 9 ;
hence not 67 divides 4967 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 4967 & n is prime holds
not n divides 4967 by XPRIMET1:38;
hence 4967 is prime by NAT_4:14; :: thesis: verum