 reserve n,s for Nat;

theorem Th74:
  for n being non zero Nat st n is even perfect holds
    n is triangular
  proof
    let n be non zero Nat;
    assume n is even perfect; then
    consider p being Nat such that
A1: 2 |^ p -' 1 is prime & n = 2 |^ (p -' 1) * (2 |^ p -' 1) by NAT_5:39;
    set p1 = Mersenne p;
A2: p <> 0
    proof
      assume p = 0; then
      2 |^ p -' 1 = 1 -' 1 by NEWTON:4
                 .= 0 by XREAL_1:232;
      hence thesis by A1;
    end;
A3: n = 2 |^ (p -' 1) * p1 by A1,XREAL_0:def 2;
A4: p -' 1 = p - 1 by XREAL_1:233,A2,NAT_1:14;
    (2 to_power p) / (2 to_power 1) = (p1 + 1) / 2; then
A5: (2 to_power (p -' 1)) = (p1 + 1) / 2 by A4,POWER:29;
    (p1 * (p1 + 1)) / 2 = Triangle p1 by Th19;
    hence thesis by A3,A5;
  end;
