defpred S₁[ natural number , natural number , set ] means $3 = F₁($1,$2);
A1: for x, y being Element of NAT ex z being Element of NAT st S₁[x,y,z]

proof

let x, y be Element of NAT ; :: thesis: ex z being Element of NAT st S₁[x,y,z]
reconsider z = F₁(x,y) as Element of NAT by ORDINAL1:def 13;
take z ; :: thesis: S₁[x,y,z]
thus S₁[x,y,z] ; :: thesis: verum

end;

consider f being Function of [:NAT ,NAT :],NAT such that
W: for x, y being Element of NAT holds S₁[x,y,f . x,y] from BINOP_1:sch 3(A1);
take f ; :: thesis: for x, y being natural number holds f . x,y = F₁(x,y)
let x, y be natural number ; :: thesis: f . x,y = F₁(x,y)
reconsider x = x, y = y as Element of NAT by ORDINAL1:def 13;
S₁[x,y,f . x,y] by W;
then f . x,y = F₁(x,y) ;
hence f . x,y = F₁(x,y) ; :: thesis: verum