reserve a,b,n for Element of NAT;

theorem Th38:
  for n being Element of NAT holds GenFib(2,1,n) = Lucas(n)
proof
  defpred P[Nat] means GenFib(2,1,$1)=Lucas($1);
A1: P[1] by Th11,Th32;
A2: for k being Nat st P[k] & P[k+1] holds P[k+2]
  proof
    let k be Nat;
    assume ( P[k])& P[k+1];
    then GenFib(2,1,k+2)= Lucas(k)+Lucas(k+1) by Th34
      .= Lucas(k+2) by Th12;
    hence thesis;
  end;
A3: P[0] by Th11,Th32;
  for k being Nat holds P[k] from FIB_NUM:sch 1 (A3, A1, A2);
  hence thesis;
end;
