reserve a,b,n for Element of NAT;

theorem Th17:
  for k being Nat holds Lucas(k) >= k
proof
  defpred P[Nat] means Lucas($1) >= $1;
A1: P[0];
A2: for k being Nat st P[k] & P[k+1] holds P[k+2]
  proof
    let k be Nat;
    assume that
A3: P[k] and
A4: P[k+1];
    per cases;
    suppose
      k = 0;
      hence thesis by Th14;
    end;
    suppose
      k <> 0;
      then
A5:   1 + (k+1) <= k + (k+1) by XREAL_1:6,NAT_1:14;
A6:   k + (k+1) <= Lucas(k) + (k+1) by A3,XREAL_1:6;
      Lucas((k + 1) + 1) = Lucas(k + 1) + Lucas(k) & Lucas(k) + (k+1) <=
      Lucas(k+1 ) + Lucas(k) by A4,Th11,XREAL_1:6;
      then k + (k+1) <= Lucas((k+1)+1) by A6,XXREAL_0:2;
      hence thesis by A5,XXREAL_0:2;
    end;
  end;
A7: P[1] by Th11;
  thus for k being Nat holds P[k] from FIB_NUM:sch 1 (A1, A7, A2);
end;
