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 Th37:
  Product(n,S) is Subset-Family of product (Seg n --> X)
  proof
    reconsider SPS=Product(n,S) as Subset of
    bool Funcs(dom (Seg n --> S),union Union (Seg n -->S)) by Th36;
    SPS c= bool product (Seg n --> X)
    proof
      let x be object;
      assume
A1:   x in SPS;
      reconsider x1=x as set by TARSKI:1;
      x1 c= product (Seg n --> X)
      proof
        let t be object;
        assume
A2:     t in x1;
        consider g be Function such that
A3:     x1 = product g and
A4:     g in product (Seg n --> S) by A1,Def2;
A5:     dom g = dom (Seg n --> S) by A4,CARD_3:9;
        consider u be Function such that
A7:     t=u and
A8:     dom u = dom g and
A9:     for v be object st v in dom g holds u.v in g.v by A2,A3,CARD_3:def 5;
        consider w be Function such that
A10:    g=w and
        dom w = dom (Seg n --> S) and
A12:    for y be object st y in dom (Seg n --> S) holds
          w.y in (Seg n --> S).y by A4,CARD_3:def 5;
A13:    dom (Seg n --> S) = Seg n & dom (Seg n --> X) = Seg n by FUNCOP_1:13;
        now
          let a be object;
          assume
A14:      a in dom (Seg n --> X);
          then reconsider a1=a as Nat;
          u.a in g.a & g.a in (Seg n --> S).a by A14,A13,A10,A12,A9,A5;
          then
A15:      u.a in union ((Seg n --> S).a) & a in Seg n
            by A14,TARSKI:def 4;
          union ((Seg n --> S).a) c= (Seg n --> X).a
          proof
A16:        (Seg n --> S).a = S & (Seg n --> X).a = X by A14,FUNCOP_1:7;
            union S c= union bool X by ZFMISC_1:77;
            hence thesis by A16,ZFMISC_1:81;
          end;
          hence u.a in (Seg n --> X).a by A15;
        end;
        hence t in product (Seg n --> X) by A13,A5,A8,A7,CARD_3:def 5;
      end;
      hence x in bool product (Seg n --> X);
    end;
    hence thesis;
  end;
