defpred S₁[ Element of NAT , set ] means $2 = (H . $1) . x;
A1: for n being Element of NAT ex y being Element of COMPLEX st S₁[n,y]

proof

let n be Element of NAT ; :: thesis: ex y being Element of COMPLEX st S₁[n,y]
(H . n) . x in COMPLEX by XCMPLX_0:def 2;
hence ex y being Element of COMPLEX st S₁[n,y] ; :: thesis: verum

end;

consider f being sequence of COMPLEX such that
A2: for n being Element of NAT holds S₁[n,f . n] from FUNCT_2:sch 3(A1);
take f ; :: thesis: for n being Nat holds f . n = (H . n) . x
let n be Nat; :: thesis: f . n = (H . n) . x
n in NAT by ORDINAL1:def 12;
hence f . n = (H . n) . x by A2; :: thesis: verum