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

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
A2: Fib k >= 0 by NAT_1:2;
Fib ((k + 1) + 1) = (Fib (k + 1)) + (Fib k) by PRE_FF:1;
then Fib ((k + 1) + 1) >= (Fib (k + 1)) + 0 by A2, XREAL_1:8;
hence ( S₁[k] implies S₁[k + 1] ) ; :: thesis: verum

end;

A3: S₁[ 0 ] by PRE_FF:1;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A3, A1); :: thesis: verum