reserve a,b,c,k,m,n for Nat;
reserve i,j,x,y for Integer;
reserve p,q for Prime;
reserve r,s for Real;

theorem
  for n being positive Nat
  ex x being Integer, y being Nat st Sum powersFS(x,3,n) = y|^3
  proof
    let n be positive Nat;
    per cases;
    suppose n is even;
      then consider k such that
A1:   n = 2*k;
      take -k,k;
      thus thesis by A1,Th49;
    end;
    suppose n is odd;
      then consider k such that
A2:   n = 2*k+1 by ABIAN:9;
      take -(k+1),0;
      n = 2*(k+1)-1 by A2;
      hence thesis by Th50;
    end;
  end;
