defpred S1[ set , set ] means $2 = n |-> (f . $1);
A1: for x being Element of T ex y being Element of (TOP-REAL n) st S1[x,y]
proof
let x be Element of T; :: thesis: ex y being Element of (TOP-REAL n) st S1[x,y]
f . x in REAL by XREAL_0:def 1;
then n |-> (f . x) is Element of REAL n by FINSEQ_2:112;
then reconsider y = n |-> (f . x) as Point of (TOP-REAL n) by EUCLID:22;
take y ; :: thesis: S1[x,y]
thus S1[x,y] ; :: thesis: verum
end;
ex F being Function of T,(TOP-REAL n) st
for x being Element of T holds S1[x,F . x] from FUNCT_2:sch 3(A1);
 hence ex b1 being Function of T,(TOP-REAL n) st
for t being Element of T holds b1 . t = n |-> (f . t) ; :: thesis: verum