now :: thesis: ( not 2 divides 1973 & not 3 divides 1973 & not 5 divides 1973 & not 7 divides 1973 & not 11 divides 1973 & not 13 divides 1973 & not 17 divides 1973 & not 19 divides 1973 & not 23 divides 1973 & not 29 divides 1973 & not 31 divides 1973 & not 37 divides 1973 & not 41 divides 1973 & not 43 divides 1973 )

1973 = (2 * 986) + 1 ;
hence not 2 divides 1973 by NAT_4:9; :: thesis: ( not 3 divides 1973 & not 5 divides 1973 & not 7 divides 1973 & not 11 divides 1973 & not 13 divides 1973 & not 17 divides 1973 & not 19 divides 1973 & not 23 divides 1973 & not 29 divides 1973 & not 31 divides 1973 & not 37 divides 1973 & not 41 divides 1973 & not 43 divides 1973 )
1973 = (3 * 657) + 2 ;
hence not 3 divides 1973 by NAT_4:9; :: thesis: ( not 5 divides 1973 & not 7 divides 1973 & not 11 divides 1973 & not 13 divides 1973 & not 17 divides 1973 & not 19 divides 1973 & not 23 divides 1973 & not 29 divides 1973 & not 31 divides 1973 & not 37 divides 1973 & not 41 divides 1973 & not 43 divides 1973 )
1973 = (5 * 394) + 3 ;
hence not 5 divides 1973 by NAT_4:9; :: thesis: ( not 7 divides 1973 & not 11 divides 1973 & not 13 divides 1973 & not 17 divides 1973 & not 19 divides 1973 & not 23 divides 1973 & not 29 divides 1973 & not 31 divides 1973 & not 37 divides 1973 & not 41 divides 1973 & not 43 divides 1973 )
1973 = (7 * 281) + 6 ;
hence not 7 divides 1973 by NAT_4:9; :: thesis: ( not 11 divides 1973 & not 13 divides 1973 & not 17 divides 1973 & not 19 divides 1973 & not 23 divides 1973 & not 29 divides 1973 & not 31 divides 1973 & not 37 divides 1973 & not 41 divides 1973 & not 43 divides 1973 )
1973 = (11 * 179) + 4 ;
hence not 11 divides 1973 by NAT_4:9; :: thesis: ( not 13 divides 1973 & not 17 divides 1973 & not 19 divides 1973 & not 23 divides 1973 & not 29 divides 1973 & not 31 divides 1973 & not 37 divides 1973 & not 41 divides 1973 & not 43 divides 1973 )
1973 = (13 * 151) + 10 ;
hence not 13 divides 1973 by NAT_4:9; :: thesis: ( not 17 divides 1973 & not 19 divides 1973 & not 23 divides 1973 & not 29 divides 1973 & not 31 divides 1973 & not 37 divides 1973 & not 41 divides 1973 & not 43 divides 1973 )
1973 = (17 * 116) + 1 ;
hence not 17 divides 1973 by NAT_4:9; :: thesis: ( not 19 divides 1973 & not 23 divides 1973 & not 29 divides 1973 & not 31 divides 1973 & not 37 divides 1973 & not 41 divides 1973 & not 43 divides 1973 )
1973 = (19 * 103) + 16 ;
hence not 19 divides 1973 by NAT_4:9; :: thesis: ( not 23 divides 1973 & not 29 divides 1973 & not 31 divides 1973 & not 37 divides 1973 & not 41 divides 1973 & not 43 divides 1973 )
1973 = (23 * 85) + 18 ;
hence not 23 divides 1973 by NAT_4:9; :: thesis: ( not 29 divides 1973 & not 31 divides 1973 & not 37 divides 1973 & not 41 divides 1973 & not 43 divides 1973 )
1973 = (29 * 68) + 1 ;
hence not 29 divides 1973 by NAT_4:9; :: thesis: ( not 31 divides 1973 & not 37 divides 1973 & not 41 divides 1973 & not 43 divides 1973 )
1973 = (31 * 63) + 20 ;
hence not 31 divides 1973 by NAT_4:9; :: thesis: ( not 37 divides 1973 & not 41 divides 1973 & not 43 divides 1973 )
1973 = (37 * 53) + 12 ;
hence not 37 divides 1973 by NAT_4:9; :: thesis: ( not 41 divides 1973 & not 43 divides 1973 )
1973 = (41 * 48) + 5 ;
hence not 41 divides 1973 by NAT_4:9; :: thesis: not 43 divides 1973
1973 = (43 * 45) + 38 ;
hence not 43 divides 1973 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 1973 & n is prime holds
not n divides 1973 by XPRIMET1:28;
hence 1973 is prime by NAT_4:14; :: thesis: verum