deffunc H₁( set , Element of [:NAT,NAT:]) -> Element of [:NAT,NAT:] = In ([($2 `2),(($2 `1) + ($2 `2))],[:NAT,NAT:]);
consider fib being sequence of [:NAT,NAT:] such that
A1: ( fib . 0 = [0,1] & ( for n being Nat holds fib . (n + 1) = H<sub>1</sub>(n,fib . n) ) ) from NAT_1:sch 12();
reconsider IT = (fib . n) `1 as Element of NAT ;
take IT ; :: thesis: ex fib being sequence of [:NAT,NAT:] st
( IT = (fib . n) `1 & fib . 0 = [0,1] & ( for n being Nat holds fib . (n + 1) = [((fib . n) `2),(((fib . n) `1) + ((fib . n) `2))] ) )
take fib ; :: thesis: ( IT = (fib . n) `1 & fib . 0 = [0,1] & ( for n being Nat holds fib . (n + 1) = [((fib . n) `2),(((fib . n) `1) + ((fib . n) `2))] ) )
thus IT = (fib . n) `1 ; :: thesis: ( fib . 0 = [0,1] & ( for n being Nat holds fib . (n + 1) = [((fib . n) `2),(((fib . n) `1) + ((fib . n) `2))] ) )
thus fib . 0 = [0,1] by A1; :: thesis: for n being Nat holds fib . (n + 1) = [((fib . n) `2),(((fib . n) `1) + ((fib . n) `2))]
let n be Nat; :: thesis: fib . (n + 1) = [((fib . n) `2),(((fib . n) `1) + ((fib . n) `2))]
fib . (n + 1) = H₁(n,fib . n) by A1;
hence fib . (n + 1) = [((fib . n) `2),(((fib . n) `1) + ((fib . n) `2))] ; :: thesis: verum