deffunc H_{1}( Element of NAT ) -> Element of the carrier of RNS = (a . $1) * (S . $1);

consider S being sequence of RNS such that

A1: for n being Element of NAT holds S . n = H_{1}(n)
from FUNCT_2:sch 4();

take S ; :: thesis: for n being Nat holds S . n = (a . n) * (S . n)

let n be Nat; :: thesis: S . n = (a . n) * (S . n)

n in NAT by ORDINAL1:def 12;

hence S . n = (a . n) * (S . n) by A1; :: thesis: verum

consider S being sequence of RNS such that

A1: for n being Element of NAT holds S . n = H

take S ; :: thesis: for n being Nat holds S . n = (a . n) * (S . n)

let n be Nat; :: thesis: S . n = (a . n) * (S . n)

n in NAT by ORDINAL1:def 12;

hence S . n = (a . n) * (S . n) by A1; :: thesis: verum