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
  (curry f in Funcs(X,Funcs(Y,Z)) or curry' f in Funcs(Y,Funcs(X,Z))) &
  dom f c= [:V1,V2:] implies f in Funcs([:X,Y:],Z)
proof
  assume curry f in Funcs(X,Funcs(Y,Z)) or curry' f in Funcs(Y,Funcs(X,Z));
  then
A1: uncurry curry f in Funcs([:X,Y:],Z) or uncurry' curry' f in Funcs([:X,Y
  :],Z) by Th11;
  assume dom f c= [:V1,V2:];
  hence thesis by A1,FUNCT_5:50;
end;
