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 LM4:
  for E,F,G be non empty set,
          f be Function of [:E,F:],G,
       x,y be object
  st x in E & y in F holds
    ( (curry f).x ).y = f.(x,y)
  proof
    let E,F,G be non empty set,
            f be Function of [:E,F:],G,
         x,y be object;
    assume that
    A1: x in E and
    A2: y in F;
    dom f = [:E,F:] by FUNCT_2:def 1; then
    ex g being Function st (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;
    hence ( (curry f).x ).y = f.(x,y) by A2;
  end;
