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 Th1:
  product f c= Funcs(dom f, Union f)
proof
  let x be object;
  assume x in product f;
  then consider g such that
A1: x = g and
A2: dom g = dom f and
A3: for x being object st x in dom f holds g.x in f.x by CARD_3:def 5;
  rng g c= Union f
  proof
    let y be object;
A4: Union f = union rng f by CARD_3:def 4;
    assume y in rng g;
    then consider z being object such that
A5: z in dom g & y = g.z by FUNCT_1:def 3;
    y in f.z & f.z in rng f by A2,A3,A5,FUNCT_1:def 3;
    hence thesis by A4,TARSKI:def 4;
  end;
  hence thesis by A1,A2,FUNCT_2:def 2;
end;
