reserve a,b,n for Element of NAT;

theorem Th33:
  for k being Nat holds (GenFib(a,b,k+1)+GenFib(a,b,(k+1)+1)) |^ 2
= (GenFib(a,b,k+1)|^2+2*GenFib(a,b,k+1) * GenFib(a,b,(k+1)+1)+GenFib(a,b,(k+1)+
  1)|^2)
proof
  let k be Nat;
  set a1 = GenFib(a,b,k+1);
  set b1 = GenFib(a,b,(k+1)+1);
  (GenFib(a,b,k+1)+GenFib(a,b,(k+1)+1))|^2 =a1*a1+a1*b1+b1*a1+b1*b1 by Th5
    .=a1*a1+2*(a1*b1)+b1*b1
    .=a1|^2+2*a1*b1+b1*b1 by WSIERP_1:1
    .=a1|^2+2*a1*b1+b1|^2 by WSIERP_1:1;
  hence thesis;
end;
