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
  product Funcs(X,f),Funcs(X, product f) are_equipotent
proof
A1: Funcs(X, product {}) = {X --> {}} by CARD_3:10,FUNCT_5:59;
A2: product Funcs({},f) = product (dom f --> {{}}) by Th45
    .= Funcs(dom f, {{}}) by CARD_3:11
    .= {dom f --> {}} by FUNCT_5:59;
A3: Funcs({} qua set, product f) = {{}} & product Funcs(X,{}) = {{}} by Th46,
CARD_3:10,FUNCT_5:57;
  X <> {} & f <> {} implies thesis by Lm4;
  hence thesis by A2,A3,A1,CARD_1:28;
end;
