reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem
  for E,F,G be non empty set,
          f be Function of [:E,F:],G,
          y be object st y in F
  holds
    (curry' f).y is Function of E,G
  proof
    let E,F,G be non empty set,
            f be Function of [:E,F:],G,
            y be object;
    assume
    A1: y in F;
    dom f = [:E,F:] by FUNCT_2:def 1; then
    ex g being Function st (curry' f) . y = g & dom g = E
      & rng g c= rng f & for x being object st x in E holds
        g . x = f . (x,y) by A1,FUNCT_5:32,ZFMISC_1:90;
    hence (curry' f) . y is Function of E,G by XBOOLE_1:1,FUNCT_2:2;
  end;
