now :: thesis: ( not 2 divides 139 & not 3 divides 139 & not 5 divides 139 & not 7 divides 139 & not 11 divides 139 )

139 = (2 * 69) + 1 ;
hence not 2 divides 139 by NAT_4:9; :: thesis: ( not 3 divides 139 & not 5 divides 139 & not 7 divides 139 & not 11 divides 139 )
139 = (3 * 46) + 1 ;
hence not 3 divides 139 by NAT_4:9; :: thesis: ( not 5 divides 139 & not 7 divides 139 & not 11 divides 139 )
139 = (5 * 27) + 4 ;
hence not 5 divides 139 by NAT_4:9; :: thesis: ( not 7 divides 139 & not 11 divides 139 )
139 = (7 * 19) + 6 ;
hence not 7 divides 139 by NAT_4:9; :: thesis: not 11 divides 139
139 = (11 * 12) + 7 ;
hence not 11 divides 139 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 139 & n is prime holds
not n divides 139 by XPRIMET1:10;
hence 139 is prime by NAT_4:14; :: thesis: verum