 reserve n,s for Nat;

theorem
  for k being Nat st k <= n holds
    Triangle k <= Triangle n
  proof
    let k be Nat;
    assume k <= n; then
    consider i being Nat such that
A1: n = k + i by NAT_1:10;
    defpred P[Nat] means
      for n being Nat holds Triangle n <= Triangle (n + $1);
A2: P[0];
A3: for k being Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume
A4:   P[k];
      let n be Nat;
A5:   Triangle n <= Triangle (n + k) by A4;
      Triangle (n + k) <= Triangle (n + k + 1) by Th42;
      hence thesis by A5,XXREAL_0:2;
    end;
    for n being Nat holds P[n] from NAT_1:sch 2(A2,A3);
    hence thesis by A1;
  end;
