reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;

theorem Th68:
  for a,b,m be odd Nat holds 4 divides a|^m + b|^m iff 4 divides a + b
  proof
    let a,b,m be odd Nat;
    consider n such that
    L0: m = 2*n+1 by ABIAN:9;
    4 divides a|^(2*n+1)+b|^(2*n+1) implies 4 divides a + b
    proof
      B2: 2|^2 = 2*2 by NEWTON:81;
      A1: 4 divides a|^(2*n) - b|^(2*n) by Th65;
      A2: 4 divides (a|^(2*n) - b|^(2*n))*((a|^1 - b|^1)/2) by Th65,INT_2:2;
      consider l such that
      A3: l = (a|^(2*n)+b|^(2*n))/2 & l is odd by A1,Th58,Lm63;
      A4: a|^(2*n+1)+b|^(2*n+1)=((a|^(2*n)+b|^(2*n))*(a|^1+b|^1)+
      (a|^(2*n)-b|^(2*n))*(a|^1-b|^1))/2 by NEWTON01:8
      .= (a|^(2*n)+b|^(2*n))*(a|^1+b|^1)/2 +
      (a|^(2*n)-b|^(2*n))*(a|^1-b|^1)/2;
      assume 4 divides a|^(2*n+1)+b|^(2*n+1); then
      2|^2 divides ((a|^(2*n) + b|^(2*n))/2)*(a|^1+b|^1) by B2,A2,A4,INT_2:1;
      hence thesis by B2,A3,Th67;
    end;
    hence thesis by L0,Lm60;
  end;
