now :: thesis: ( not 2 divides 2063 & not 3 divides 2063 & not 5 divides 2063 & not 7 divides 2063 & not 11 divides 2063 & not 13 divides 2063 & not 17 divides 2063 & not 19 divides 2063 & not 23 divides 2063 & not 29 divides 2063 & not 31 divides 2063 & not 37 divides 2063 & not 41 divides 2063 & not 43 divides 2063 )

2063 = (2 * 1031) + 1 ;
hence not 2 divides 2063 by NAT_4:9; :: thesis: ( not 3 divides 2063 & not 5 divides 2063 & not 7 divides 2063 & not 11 divides 2063 & not 13 divides 2063 & not 17 divides 2063 & not 19 divides 2063 & not 23 divides 2063 & not 29 divides 2063 & not 31 divides 2063 & not 37 divides 2063 & not 41 divides 2063 & not 43 divides 2063 )
2063 = (3 * 687) + 2 ;
hence not 3 divides 2063 by NAT_4:9; :: thesis: ( not 5 divides 2063 & not 7 divides 2063 & not 11 divides 2063 & not 13 divides 2063 & not 17 divides 2063 & not 19 divides 2063 & not 23 divides 2063 & not 29 divides 2063 & not 31 divides 2063 & not 37 divides 2063 & not 41 divides 2063 & not 43 divides 2063 )
2063 = (5 * 412) + 3 ;
hence not 5 divides 2063 by NAT_4:9; :: thesis: ( not 7 divides 2063 & not 11 divides 2063 & not 13 divides 2063 & not 17 divides 2063 & not 19 divides 2063 & not 23 divides 2063 & not 29 divides 2063 & not 31 divides 2063 & not 37 divides 2063 & not 41 divides 2063 & not 43 divides 2063 )
2063 = (7 * 294) + 5 ;
hence not 7 divides 2063 by NAT_4:9; :: thesis: ( not 11 divides 2063 & not 13 divides 2063 & not 17 divides 2063 & not 19 divides 2063 & not 23 divides 2063 & not 29 divides 2063 & not 31 divides 2063 & not 37 divides 2063 & not 41 divides 2063 & not 43 divides 2063 )
2063 = (11 * 187) + 6 ;
hence not 11 divides 2063 by NAT_4:9; :: thesis: ( not 13 divides 2063 & not 17 divides 2063 & not 19 divides 2063 & not 23 divides 2063 & not 29 divides 2063 & not 31 divides 2063 & not 37 divides 2063 & not 41 divides 2063 & not 43 divides 2063 )
2063 = (13 * 158) + 9 ;
hence not 13 divides 2063 by NAT_4:9; :: thesis: ( not 17 divides 2063 & not 19 divides 2063 & not 23 divides 2063 & not 29 divides 2063 & not 31 divides 2063 & not 37 divides 2063 & not 41 divides 2063 & not 43 divides 2063 )
2063 = (17 * 121) + 6 ;
hence not 17 divides 2063 by NAT_4:9; :: thesis: ( not 19 divides 2063 & not 23 divides 2063 & not 29 divides 2063 & not 31 divides 2063 & not 37 divides 2063 & not 41 divides 2063 & not 43 divides 2063 )
2063 = (19 * 108) + 11 ;
hence not 19 divides 2063 by NAT_4:9; :: thesis: ( not 23 divides 2063 & not 29 divides 2063 & not 31 divides 2063 & not 37 divides 2063 & not 41 divides 2063 & not 43 divides 2063 )
2063 = (23 * 89) + 16 ;
hence not 23 divides 2063 by NAT_4:9; :: thesis: ( not 29 divides 2063 & not 31 divides 2063 & not 37 divides 2063 & not 41 divides 2063 & not 43 divides 2063 )
2063 = (29 * 71) + 4 ;
hence not 29 divides 2063 by NAT_4:9; :: thesis: ( not 31 divides 2063 & not 37 divides 2063 & not 41 divides 2063 & not 43 divides 2063 )
2063 = (31 * 66) + 17 ;
hence not 31 divides 2063 by NAT_4:9; :: thesis: ( not 37 divides 2063 & not 41 divides 2063 & not 43 divides 2063 )
2063 = (37 * 55) + 28 ;
hence not 37 divides 2063 by NAT_4:9; :: thesis: ( not 41 divides 2063 & not 43 divides 2063 )
2063 = (41 * 50) + 13 ;
hence not 41 divides 2063 by NAT_4:9; :: thesis: not 43 divides 2063
2063 = (43 * 47) + 42 ;
hence not 43 divides 2063 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 2063 & n is prime holds
not n divides 2063 by XPRIMET1:28;
hence 2063 is prime by NAT_4:14; :: thesis: verum