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,
          x be object st x in E
  holds
    (curry f).x is Function of F,G
  proof
    let E,F,G be non empty set,
            f be Function of [:E,F:],G,
            x be object;
    assume
    A1: x in E;
    dom f = [:E,F:] by FUNCT_2:def 1; then
    consider g being Function such that
    A4: (curry f) . x = g
      & dom g = F
      & rng g c= rng f
      & for y being object st y in F holds
        g . y = f . (x,y) by A1,FUNCT_5:29,ZFMISC_1:90;
    thus (curry f) . x is Function of F,G by A4,XBOOLE_1:1,FUNCT_2:2;
  end;
