now :: thesis: ( not 2 divides 2143 & not 3 divides 2143 & not 5 divides 2143 & not 7 divides 2143 & not 11 divides 2143 & not 13 divides 2143 & not 17 divides 2143 & not 19 divides 2143 & not 23 divides 2143 & not 29 divides 2143 & not 31 divides 2143 & not 37 divides 2143 & not 41 divides 2143 & not 43 divides 2143 )

2143 = (2 * 1071) + 1 ;
hence not 2 divides 2143 by NAT_4:9; :: thesis: ( not 3 divides 2143 & not 5 divides 2143 & not 7 divides 2143 & not 11 divides 2143 & not 13 divides 2143 & not 17 divides 2143 & not 19 divides 2143 & not 23 divides 2143 & not 29 divides 2143 & not 31 divides 2143 & not 37 divides 2143 & not 41 divides 2143 & not 43 divides 2143 )
2143 = (3 * 714) + 1 ;
hence not 3 divides 2143 by NAT_4:9; :: thesis: ( not 5 divides 2143 & not 7 divides 2143 & not 11 divides 2143 & not 13 divides 2143 & not 17 divides 2143 & not 19 divides 2143 & not 23 divides 2143 & not 29 divides 2143 & not 31 divides 2143 & not 37 divides 2143 & not 41 divides 2143 & not 43 divides 2143 )
2143 = (5 * 428) + 3 ;
hence not 5 divides 2143 by NAT_4:9; :: thesis: ( not 7 divides 2143 & not 11 divides 2143 & not 13 divides 2143 & not 17 divides 2143 & not 19 divides 2143 & not 23 divides 2143 & not 29 divides 2143 & not 31 divides 2143 & not 37 divides 2143 & not 41 divides 2143 & not 43 divides 2143 )
2143 = (7 * 306) + 1 ;
hence not 7 divides 2143 by NAT_4:9; :: thesis: ( not 11 divides 2143 & not 13 divides 2143 & not 17 divides 2143 & not 19 divides 2143 & not 23 divides 2143 & not 29 divides 2143 & not 31 divides 2143 & not 37 divides 2143 & not 41 divides 2143 & not 43 divides 2143 )
2143 = (11 * 194) + 9 ;
hence not 11 divides 2143 by NAT_4:9; :: thesis: ( not 13 divides 2143 & not 17 divides 2143 & not 19 divides 2143 & not 23 divides 2143 & not 29 divides 2143 & not 31 divides 2143 & not 37 divides 2143 & not 41 divides 2143 & not 43 divides 2143 )
2143 = (13 * 164) + 11 ;
hence not 13 divides 2143 by NAT_4:9; :: thesis: ( not 17 divides 2143 & not 19 divides 2143 & not 23 divides 2143 & not 29 divides 2143 & not 31 divides 2143 & not 37 divides 2143 & not 41 divides 2143 & not 43 divides 2143 )
2143 = (17 * 126) + 1 ;
hence not 17 divides 2143 by NAT_4:9; :: thesis: ( not 19 divides 2143 & not 23 divides 2143 & not 29 divides 2143 & not 31 divides 2143 & not 37 divides 2143 & not 41 divides 2143 & not 43 divides 2143 )
2143 = (19 * 112) + 15 ;
hence not 19 divides 2143 by NAT_4:9; :: thesis: ( not 23 divides 2143 & not 29 divides 2143 & not 31 divides 2143 & not 37 divides 2143 & not 41 divides 2143 & not 43 divides 2143 )
2143 = (23 * 93) + 4 ;
hence not 23 divides 2143 by NAT_4:9; :: thesis: ( not 29 divides 2143 & not 31 divides 2143 & not 37 divides 2143 & not 41 divides 2143 & not 43 divides 2143 )
2143 = (29 * 73) + 26 ;
hence not 29 divides 2143 by NAT_4:9; :: thesis: ( not 31 divides 2143 & not 37 divides 2143 & not 41 divides 2143 & not 43 divides 2143 )
2143 = (31 * 69) + 4 ;
hence not 31 divides 2143 by NAT_4:9; :: thesis: ( not 37 divides 2143 & not 41 divides 2143 & not 43 divides 2143 )
2143 = (37 * 57) + 34 ;
hence not 37 divides 2143 by NAT_4:9; :: thesis: ( not 41 divides 2143 & not 43 divides 2143 )
2143 = (41 * 52) + 11 ;
hence not 41 divides 2143 by NAT_4:9; :: thesis: not 43 divides 2143
2143 = (43 * 49) + 36 ;
hence not 43 divides 2143 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 2143 & n is prime holds
not n divides 2143 by XPRIMET1:28;
hence 2143 is prime by NAT_4:14; :: thesis: verum