now :: thesis: ( not 2 divides 61 & not 3 divides 61 & not 5 divides 61 & not 7 divides 61 )

61 = (2 * 30) + 1 ;
hence not 2 divides 61 by NAT_4:9; :: thesis: ( not 3 divides 61 & not 5 divides 61 & not 7 divides 61 )
61 = (3 * 20) + 1 ;
hence not 3 divides 61 by NAT_4:9; :: thesis: ( not 5 divides 61 & not 7 divides 61 )
61 = (5 * 12) + 1 ;
hence not 5 divides 61 by NAT_4:9; :: thesis: not 7 divides 61
61 = (7 * 8) + 5 ;
hence not 7 divides 61 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 61 & n is prime holds
not n divides 61 by XPRIMET1:8;
hence 61 is prime by NAT_4:14; :: thesis: verum