 reserve n,s for Nat;

theorem Th45:
  for n being non trivial Nat holds
    n < Triangle n
  proof
    let n be non trivial Nat;
    defpred P[Nat] means $1 < Triangle $1;
A1: P[2] by Th12;
A2: for k being non trivial Nat st P[k] holds P[k+1]
    proof
      let k be non trivial Nat;
      assume P[k];
      Triangle (k + 1) = Triangle (k) + (k + 1) by Th10;
      hence thesis by NAT_1:16;
    end;
    for n being non trivial Nat holds P[n] from NAT_2:sch 2(A1,A2);
    hence thesis;
  end;
