deffunc H₁( Nat, Element of COMPLEX ) -> Element of COMPLEX = $2 * ($1 + 1);
consider f being Function of NAT ,COMPLEX such that
A1: ( f . 0 = 1r & ( for n being Nat holds f . (n + 1) = H₁(n,f . n) ) ) from NAT_1:sch 12();
take f ; :: thesis: ( f . 0 = 1 & ( for n being Element of NAT holds f . (n + 1) = (f . n) * (n + 1) ) )
thus f . 0 = 1 by A1; :: thesis: for n being Element of NAT holds f . (n + 1) = (f . n) * (n + 1)
let n be Element of NAT ; :: thesis: f . (n + 1) = (f . n) * (n + 1)
thus f . (n + 1) = (f . n) * (n + 1) by A1; :: thesis: verum