deffunc H_{1}( Element of Bags X) -> Element of the carrier of L = (p . $1) * a;

consider f being Function of (Bags X), the carrier of L such that

A4: for x being Element of Bags X holds f . x = H_{1}(x)
from FUNCT_2:sch 4();

reconsider f = f as Function of (Bags X),L ;

reconsider f = f as Series of X,L ;

reconsider f = f as Series of X,L ;

take f ; :: thesis: for b being bag of X holds f . b = (p . b) * a

let x be bag of X; :: thesis: f . x = (p . x) * a

x in Bags X by PRE_POLY:def 12;

hence f . x = (p . x) * a by A4; :: thesis: verum

consider f being Function of (Bags X), the carrier of L such that

A4: for x being Element of Bags X holds f . x = H

reconsider f = f as Function of (Bags X),L ;

reconsider f = f as Series of X,L ;

reconsider f = f as Series of X,L ;

take f ; :: thesis: for b being bag of X holds f . b = (p . b) * a

let x be bag of X; :: thesis: f . x = (p . x) * a

x in Bags X by PRE_POLY:def 12;

hence f . x = (p . x) * a by A4; :: thesis: verum