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 PFuncs(X,PFuncs(Y,Z)) or curry' f in PFuncs(Y,PFuncs(X,Z))
  ) & dom f c= [:V1,V2:] implies f in PFuncs([:X,Y:],Z)
proof
  assume
  curry f in PFuncs(X,PFuncs(Y,Z)) or curry' f in PFuncs(Y,PFuncs(X,Z) );
  then
A1: uncurry curry f in PFuncs([:X,Y:],Z) or uncurry' curry' f in PFuncs([:X,
  Y:],Z) by Th15;
  assume dom f c= [:V1,V2:];
  hence thesis by A1,FUNCT_5:50;
end;
