let n be non trivial Nat; :: thesis: n < Triangle n
defpred S₁[ Nat] means $1 < Triangle $1;
A1: S₁[2] by Th12;
A2: for k being non trivial Nat st S₁[k] holds
S₁[k + 1]

proof

let k be non trivial Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume S₁[k] ; :: thesis: S₁[k + 1]
Triangle (k + 1) = (Triangle k) + (k + 1) by Th10;
hence S₁[k + 1] by NAT_1:16; :: thesis: verum

end;

for n being non trivial Nat holds S₁[n] from NAT_2:sch 2(A1, A2);
hence n < Triangle n ; :: thesis: verum