reserve x,y,y1,y2,z,a,b for object, X,Y,Z,V1,V2 for set,
  f,g,h,h9,f1,f2 for Function,
  i for Nat,
  P for Permutation of X,
  D,D1,D2,D3 for non empty set,
  d1 for Element of D1,
  d2 for Element of D2,
  d3 for Element of D3;

theorem
  X <> {} implies rng uncurry (X --> f) = rng f & rng uncurry' (X --> f)
  = rng f
proof
  set x = the Element of X;
  assume
A1: X <> {};
  then dom (X --> f) = X & (X --> f).x = f by FUNCOP_1:7;
  hence rng uncurry (X --> f) c= rng f & rng f c= rng uncurry (X --> f) & rng
  uncurry' (X --> f) c= rng f & rng f c= rng uncurry' (X --> f) by A1,Th7,Th8;
end;
