reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve p for Prime;

theorem Th44:
  for n being non zero Nat holds 3 divides 2|^(2|^(n+1)) + 2|^(2|^n) + 1
  proof
    let n be non zero Nat;
A1: 2|^(2|^(n+1)) mod 3 = 1 by Th42;
A2: 2|^(2|^n) mod 3 = 1 by Th42;
    (2|^(2|^(n+1))+2|^(2|^n)+1) mod 3 = (1+1+1) mod 3 by A1,A2,Lm8,NUMBER05:8
    .= 0;
    hence thesis by INT_1:62;
  end;
