now :: thesis: ( not 2 divides 89 & not 3 divides 89 & not 5 divides 89 & not 7 divides 89 )

89 = (2 * 44) + 1 ;
hence not 2 divides 89 by NAT_4:9; :: thesis: ( not 3 divides 89 & not 5 divides 89 & not 7 divides 89 )
89 = (3 * 29) + 2 ;
hence not 3 divides 89 by NAT_4:9; :: thesis: ( not 5 divides 89 & not 7 divides 89 )
89 = (5 * 17) + 4 ;
hence not 5 divides 89 by NAT_4:9; :: thesis: not 7 divides 89
89 = (7 * 12) + 5 ;
hence not 7 divides 89 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 89 & n is prime holds
not n divides 89 by XPRIMET1:8;
hence 89 is prime by NAT_4:14; :: thesis: verum