defpred S_{1}[ Nat] means (Fib $1) gcd (Fib ($1 + 1)) = 1;

_{1}[ 0 ]
by NEWTON:52, PRE_FF:1;

thus for m being Nat holds S_{1}[m]
from NAT_1:sch 2(A3, A1); :: thesis: verum

A1: now :: thesis: for k being Nat st S_{1}[k] holds

S_{1}[k + 1]

A3:
SS

let k be Nat; :: thesis: ( S_{1}[k] implies S_{1}[k + 1] )

assume A2: S_{1}[k]
; :: thesis: S_{1}[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 S_{1}[k + 1]
; :: thesis: verum

end;assume A2: S

(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 S

thus for m being Nat holds S