deffunc H₁( Nat) -> Element of REAL = In ((Sum (Prob * (A ^\ $1))),REAL);
consider f being Real_Sequence such that
A1: for k being Element of NAT holds f . k = H₁(k) from FUNCT_2:sch 4();
take f ; :: thesis: for n being Nat holds f . n = Sum (Prob * (A ^\ n))
let k be Nat; :: thesis: f . k = Sum (Prob * (A ^\ k))
k in NAT by ORDINAL1:def 12;
then f . k = H₁(k) by A1;
hence f . k = Sum (Prob * (A ^\ k)) ; :: thesis: verum