7027 = (2 * 3513) + 1 ;
hence not 2 divides 7027 by NAT_4:9; :: thesis:
7027 = (3 * 2342) + 1 ;
hence not 3 divides 7027 by NAT_4:9; :: thesis:
7027 = (5 * 1405) + 2 ;
hence not 5 divides 7027 by NAT_4:9; :: thesis:
7027 = (7 * 1003) + 6 ;
hence not 7 divides 7027 by NAT_4:9; :: thesis:
7027 = (11 * 638) + 9 ;
hence not 11 divides 7027 by NAT_4:9; :: thesis:
7027 = (13 * 540) + 7 ;
hence not 13 divides 7027 by NAT_4:9; :: thesis:
7027 = (17 * 413) + 6 ;
hence not 17 divides 7027 by NAT_4:9; :: thesis:
7027 = (19 * 369) + 16 ;
hence not 19 divides 7027 by NAT_4:9; :: thesis:
7027 = (23 * 305) + 12 ;
hence not 23 divides 7027 by NAT_4:9; :: thesis:
7027 = (29 * 242) + 9 ;
hence not 29 divides 7027 by NAT_4:9; :: thesis:
7027 = (31 * 226) + 21 ;
hence not 31 divides 7027 by NAT_4:9; :: thesis:
7027 = (37 * 189) + 34 ;
hence not 37 divides 7027 by NAT_4:9; :: thesis:
7027 = (41 * 171) + 16 ;
hence not 41 divides 7027 by NAT_4:9; :: thesis:
7027 = (43 * 163) + 18 ;
hence not 43 divides 7027 by NAT_4:9; :: thesis:
7027 = (47 * 149) + 24 ;
hence not 47 divides 7027 by NAT_4:9; :: thesis:
7027 = (53 * 132) + 31 ;
hence not 53 divides 7027 by NAT_4:9; :: thesis:
7027 = (59 * 119) + 6 ;
hence not 59 divides 7027 by NAT_4:9; :: thesis:
7027 = (61 * 115) + 12 ;
hence not 61 divides 7027 by NAT_4:9; :: thesis:
7027 = (67 * 104) + 59 ;
hence not 67 divides 7027 by NAT_4:9; :: thesis:
7027 = (71 * 98) + 69 ;
hence not 71 divides 7027 by NAT_4:9; :: thesis:
7027 = (73 * 96) + 19 ;
hence not 73 divides 7027 by NAT_4:9; :: thesis:
7027 = (79 * 88) + 75 ;
hence not 79 divides 7027 by NAT_4:9; :: thesis:
7027 = (83 * 84) + 55 ;
hence not 83 divides 7027 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7027 & n is prime holds
not n divides 7027 by XPRIMET1:46;
hence 7027 is prime by NAT_4:14; :: thesis: