reserve n   for Nat,
        r,s for Real,
        x,y for Element of REAL n,
        p,q for Point of TOP-REAL n,
        e   for Point of Euclid n;
reserve n for non zero Nat;
reserve n for non zero Nat;
reserve n for Nat,
        X for set,
        S for Subset-Family of X;

theorem Th36:
  Product(n,X) c= bool Funcs(dom (Seg n --> X), union Union (Seg n --> X))
  proof
    let x be object;
    assume x in Product(n,X);
    then consider g be Function such that
A1:   x = product g and
A2:   g in product (Seg n --> X) by Def2;
A3:   dom g = dom (Seg n --> X) by A2,CARD_3:9;
    rng g c= Union (Seg n --> X)
    proof
      let t be object;
      assume t in rng g;
      then consider u be object such that
A4:     u in dom g and
A5:     t=g.u by FUNCT_1:def 3;
      consider h be Function such that
A6:   g = h and
A7:   dom h = dom (Seg n --> X) and
A8:   for v be object st v in dom (Seg n --> X) holds h.v in (Seg n --> X).v
        by A2,CARD_3:def 5;
      t in (Seg n --> X).u & (Seg n --> X).u in rng (Seg n --> X)
        by A6,A7,A8,A4,A5,FUNCT_1:def 3;
      hence thesis by TARSKI:def 4;
    end;
    then Union g c= union Union (Seg n --> X) by ZFMISC_1:77;
    then Funcs(dom g,Union g) c=
      Funcs(dom (Seg n --> X), union Union (Seg n --> X))
    by A3,FUNCT_5:56; then
A9: bool Funcs(dom g,Union g) c=
      bool Funcs(dom (Seg n --> X), union Union (Seg n --> X))
      by ZFMISC_1:67;
    product g c= Funcs(dom g, Union g) by FUNCT_6:1;
    hence thesis by A1,A9;
  end;
