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

proof

let x be Element of REAL ; :: thesis: ex y being Element of REAL st S₁[x,y]
reconsider y = F₁(x) as Element of REAL by XREAL_0:def 1;
take y ; :: thesis: S₁[x,y]
thus S₁[x,y] ; :: thesis: verum

end;

consider f being Function of REAL,REAL such that
A2: for x being Element of REAL holds S₁[x,f . x] from FUNCT_2:sch 3(A1);
take f ; :: thesis: for x being Real holds f . x = F₁(x)
let c be Real; :: thesis: f . c = F₁(c)
reconsider c = c as Element of REAL by XREAL_0:def 1;
S₁[c,f . c] by A2;
hence f . c = F₁(c) ; :: thesis: verum