defpred S₁[ Nat] means Fib ($1 + 1) >= Fib $1;
A1: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
Fib ((k + 1) + 1) = (Fib (k + 1)) + (Fib k) by PRE_FF:1;
then Fib ((k + 1) + 1) >= (Fib (k + 1)) + 0 by XREAL_1:6;
hence ( S₁[k] implies S₁[k + 1] ) ; :: thesis: verum

end;

A2: S₁[ 0 ] by PRE_FF:1;
thus for k being Nat holds S₁[k] from NAT_1:sch 2(A2, A1); :: thesis: verum