now :: thesis: ( not 2 divides 1993 & not 3 divides 1993 & not 5 divides 1993 & not 7 divides 1993 & not 11 divides 1993 & not 13 divides 1993 & not 17 divides 1993 & not 19 divides 1993 & not 23 divides 1993 & not 29 divides 1993 & not 31 divides 1993 & not 37 divides 1993 & not 41 divides 1993 & not 43 divides 1993 )

1993 = (2 * 996) + 1 ;
hence not 2 divides 1993 by NAT_4:9; :: thesis: ( not 3 divides 1993 & not 5 divides 1993 & not 7 divides 1993 & not 11 divides 1993 & not 13 divides 1993 & not 17 divides 1993 & not 19 divides 1993 & not 23 divides 1993 & not 29 divides 1993 & not 31 divides 1993 & not 37 divides 1993 & not 41 divides 1993 & not 43 divides 1993 )
1993 = (3 * 664) + 1 ;
hence not 3 divides 1993 by NAT_4:9; :: thesis: ( not 5 divides 1993 & not 7 divides 1993 & not 11 divides 1993 & not 13 divides 1993 & not 17 divides 1993 & not 19 divides 1993 & not 23 divides 1993 & not 29 divides 1993 & not 31 divides 1993 & not 37 divides 1993 & not 41 divides 1993 & not 43 divides 1993 )
1993 = (5 * 398) + 3 ;
hence not 5 divides 1993 by NAT_4:9; :: thesis: ( not 7 divides 1993 & not 11 divides 1993 & not 13 divides 1993 & not 17 divides 1993 & not 19 divides 1993 & not 23 divides 1993 & not 29 divides 1993 & not 31 divides 1993 & not 37 divides 1993 & not 41 divides 1993 & not 43 divides 1993 )
1993 = (7 * 284) + 5 ;
hence not 7 divides 1993 by NAT_4:9; :: thesis: ( not 11 divides 1993 & not 13 divides 1993 & not 17 divides 1993 & not 19 divides 1993 & not 23 divides 1993 & not 29 divides 1993 & not 31 divides 1993 & not 37 divides 1993 & not 41 divides 1993 & not 43 divides 1993 )
1993 = (11 * 181) + 2 ;
hence not 11 divides 1993 by NAT_4:9; :: thesis: ( not 13 divides 1993 & not 17 divides 1993 & not 19 divides 1993 & not 23 divides 1993 & not 29 divides 1993 & not 31 divides 1993 & not 37 divides 1993 & not 41 divides 1993 & not 43 divides 1993 )
1993 = (13 * 153) + 4 ;
hence not 13 divides 1993 by NAT_4:9; :: thesis: ( not 17 divides 1993 & not 19 divides 1993 & not 23 divides 1993 & not 29 divides 1993 & not 31 divides 1993 & not 37 divides 1993 & not 41 divides 1993 & not 43 divides 1993 )
1993 = (17 * 117) + 4 ;
hence not 17 divides 1993 by NAT_4:9; :: thesis: ( not 19 divides 1993 & not 23 divides 1993 & not 29 divides 1993 & not 31 divides 1993 & not 37 divides 1993 & not 41 divides 1993 & not 43 divides 1993 )
1993 = (19 * 104) + 17 ;
hence not 19 divides 1993 by NAT_4:9; :: thesis: ( not 23 divides 1993 & not 29 divides 1993 & not 31 divides 1993 & not 37 divides 1993 & not 41 divides 1993 & not 43 divides 1993 )
1993 = (23 * 86) + 15 ;
hence not 23 divides 1993 by NAT_4:9; :: thesis: ( not 29 divides 1993 & not 31 divides 1993 & not 37 divides 1993 & not 41 divides 1993 & not 43 divides 1993 )
1993 = (29 * 68) + 21 ;
hence not 29 divides 1993 by NAT_4:9; :: thesis: ( not 31 divides 1993 & not 37 divides 1993 & not 41 divides 1993 & not 43 divides 1993 )
1993 = (31 * 64) + 9 ;
hence not 31 divides 1993 by NAT_4:9; :: thesis: ( not 37 divides 1993 & not 41 divides 1993 & not 43 divides 1993 )
1993 = (37 * 53) + 32 ;
hence not 37 divides 1993 by NAT_4:9; :: thesis: ( not 41 divides 1993 & not 43 divides 1993 )
1993 = (41 * 48) + 25 ;
hence not 41 divides 1993 by NAT_4:9; :: thesis: not 43 divides 1993
1993 = (43 * 46) + 15 ;
hence not 43 divides 1993 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 1993 & n is prime holds
not n divides 1993 by XPRIMET1:28;
hence 1993 is prime by NAT_4:14; :: thesis: verum