defpred S₁[ Nat, object ] means $2 = ((- 1) |^ $1) * (a . $1);
A1: for x being Element of NAT ex y being Element of REAL st S₁[x,y]

proof

let x be Element of NAT ; :: thesis: ex y being Element of REAL st S₁[x,y]
((- 1) |^ x) * (a . x) in REAL by XREAL_0:def 1;
hence ex y being Element of REAL st S₁[x,y] ; :: thesis: verum

end;

consider f being Function of NAT,REAL such that
A2: for x being Element of NAT holds S₁[x,f . x] from FUNCT_2:sch 3(A1);
take f ; :: thesis: for i being Nat holds f . i = ((- 1) |^ i) * (a . i)
let i be Nat; :: thesis: f . i = ((- 1) |^ i) * (a . i)
i in NAT by ORDINAL1:def 12;
hence f . i = ((- 1) |^ i) * (a . i) by A2; :: thesis: verum