reserve a,b,n for Element of NAT;

theorem
  for m being non zero Element of NAT holds Lucas(m+1) >= Lucas(m)
proof
  let m be non zero Element of NAT;
  thus Lucas(m+1)>=Lucas(m)
  proof
    defpred P[Nat] means Lucas($1+1) >= Lucas($1);
A1: for k being non zero Nat st P[k] & P[k+1] holds P[k+2]
    proof
      let k be non zero Nat;
      assume that
      P[k] and
      P[k+1];
      Lucas(k+2)+Lucas(k+1)=Lucas(k+3) by Th13;
      hence P[k+2] by NAT_1:12;
    end;
A2: P[2] by Th14,Th15;
A3: P[1] by Th11,Th14;
    for k being non zero Nat holds P[k] from FIB_NUM2:sch 1(A3,A2, A1);
    hence thesis;
  end;
end;
