deffunc H₁( Nat) -> Element of the carrier of Y = (H . $1) /. x;
consider f being sequence of the carrier of Y such that
A1: for n being Element of NAT holds f . n = H₁(n) from FUNCT_2:sch 4();
reconsider f = f as sequence of the carrier of Y ;
take f ; :: thesis: for n being Nat holds f . n = (H . n) /. x
for n being Nat holds f . n = (H . n) /. x

proof

let n be Nat; :: thesis: f . n = (H . n) /. x
n is Element of NAT by ORDINAL1:def 12;
hence f . n = (H . n) /. x by A1; :: thesis: verum

end;

hence for n being Nat holds f . n = (H . n) /. x ; :: thesis: verum