A1: 2 |^ 2 = 2 |^ (1 + 1)
.= (2 |^ 1) * 2 by NEWTON:6
.= 2 * 2 ;
then 2 |^ (2 + 2) = 4 * 4 by NEWTON:8;
then A2: 2 |^ (4 + 2) = 16 * 4 by A1, NEWTON:8;
A3: 2 |^ (6 + 1) = (2 |^ 6) * (2 |^ 1) by NEWTON:8
.= 64 * 2 by A2 ;
A4: 5 = (2 * 2) + 1 ;
641 = (5 * (2 |^ 7)) + 1 by A3;
hence 641 is Proth by A3, A4; :: thesis: verum