defpred S₁[ Nat] means (Fib $1) gcd (Fib ($1 + 1)) = 1;

A1: now :: thesis: for k being Nat st S₁[k] holds
S₁[k + 1]

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A2: S₁[k] ; :: thesis: S₁[k + 1]
(Fib (k + 1)) gcd (Fib ((k + 1) + 1)) = (Fib (k + 1)) gcd ((Fib (k + 1)) + (Fib k)) by PRE_FF:1
.= 1 by A2, Th1 ;
hence S₁[k + 1] ; :: thesis: verum

end;

A3: S₁[ 0 ] by NEWTON:52, PRE_FF:1;
thus for m being Nat holds S₁[m] from NAT_1:sch 2(A3, A1); :: thesis: verum