
theorem Th34:
  for X being non empty TopSpace, M being non empty set ex F being
  Function of oContMaps(X, M-TOP_prod (M --> Sierpinski_Space)), M-POS_prod (M
--> oContMaps(X, Sierpinski_Space)) st F is isomorphic & for f being continuous
  Function of X, M-TOP_prod (M --> Sierpinski_Space) holds F.f = commute f
proof
  let X be non empty TopSpace, M be non empty set;
  set S = Sierpinski_Space, S9M = M-TOP_prod (M --> S);
  set XxxS9M = oContMaps(X, S9M), XxS = oContMaps(X, S);
  set XxS9xM = M-POS_prod (M --> XxS);
  deffunc F(Element of XxxS9M) = commute $1;
  consider F being ManySortedSet of the carrier of XxxS9M such that
A1: for f being Element of XxxS9M holds F.f = F(f) from PBOOLE:sch 5;
A2: dom F = the carrier of XxxS9M by PARTFUN1:def 2;
  rng F c= the carrier of XxS9xM
  proof
    let g be object;
    assume g in rng F;
    then consider f being object such that
A3: f in dom F and
A4: g = F.f by FUNCT_1:def 3;
    reconsider f as continuous Function of X, S9M by A3,Th2;
    g = commute f by A1,A3,A4;
    then reconsider g as Function of M, the carrier of XxS by Th31;
    dom g = M & rng g c= the carrier of XxS by FUNCT_2:def 1;
    then g in Funcs(M, the carrier of XxS) by FUNCT_2:def 2;
    then g in the carrier of XxS|^M by YELLOW_1:28;
    hence thesis by YELLOW_1:def 5;
  end;
  then reconsider F as Function of XxxS9M, XxS9xM by A2,FUNCT_2:2;
  deffunc F(Element of XxS9xM) = commute $1;
  consider G being ManySortedSet of the carrier of XxS9xM such that
A5: for f being Element of XxS9xM holds G.f = F(f) from PBOOLE:sch 5;
A6: dom G = the carrier of XxS9xM by PARTFUN1:def 2;
A7: rng G c= the carrier of XxxS9M
  proof
    let g be object;
    assume g in rng G;
    then consider f being object such that
A8: f in dom G and
A9: g = G.f by FUNCT_1:def 3;
    f in the carrier of XxS|^M by A6,A8,YELLOW_1:def 5;
    then f in Funcs(M, the carrier of XxS) by YELLOW_1:28;
    then consider f9 being Function such that
A10: f = f9 and
A11: dom f9 = M & rng f9 c= the carrier of XxS by FUNCT_2:def 2;
A12: f9 is Function of M, the carrier of XxS by A11,FUNCT_2:2;
    g = commute f9 by A5,A8,A9,A10;
    then g is continuous Function of X, S9M by A12,Th33;
    then g is Element of XxxS9M by Th2;
    hence thesis;
  end;
  take F;
A13: the carrier of S9M = product Carrier (M --> S) by WAYBEL18:def 3
    .= product (M --> the carrier of S) by Th30
    .= Funcs(M, the carrier of S) by CARD_3:11;
  reconsider G as Function of XxS9xM, XxxS9M by A6,A7,FUNCT_2:2;
A14: the carrier of XxxS9M c= Funcs(the carrier of X, the carrier of S9M) by
Th32;
  now
    let a be Element of XxxS9M;
A15: commute commute a = a by A14,A13,FUNCT_6:57;
    thus (G*F).a = G.(F.a) by FUNCT_2:15
      .= commute (F.a) by A5
      .= a by A1,A15
      .= (id XxxS9M).a;
  end;
  then
A16: G*F = id XxxS9M by FUNCT_2:63;
A17: the RelStr of Omega S9M = M-POS_prod(M --> Omega S) by WAYBEL25:14;
A18: F is monotone
  proof
    let a, b be Element of XxxS9M such that
A19: a <= b;
    reconsider f9 = a, g9 = b as continuous Function of X, S9M by Th2;
    reconsider f = a, g = b as continuous Function of X, Omega S9M by Th1;
    now
      let i be Element of M;
A20:  (M --> XxS).i = XxS by FUNCOP_1:7;
      then reconsider
      Fai = (F.a).i, Fbi = (F.b).i as continuous Function of X,
      Omega S by Th1;
      now
        let j be set;
        assume j in the carrier of X;
        then reconsider x = j as Element of X;
        b in the carrier of XxxS9M & F.b = commute g by A1;
        then
A21:    Fbi.x = (g9.x).i by A14,A13,FUNCT_6:56;
        reconsider fx = f.x, gx = g.x as Element of M-POS_prod(M --> Omega S)
        by A17;
        a in the carrier of XxxS9M & F.a = commute f by A1;
        then
A22:    Fai.x = (f9.x).i by A14,A13,FUNCT_6:56;
        f <= g by A19,Th3;
        then ex a, b being Element of Omega S9M st a = f.x & b = g.x & a <= b;
        then fx <= gx by A17,YELLOW_0:1;
        then
A23:    fx.i <= gx.i by WAYBEL_3:28;
        (M --> Omega S).i = Omega S by FUNCOP_1:7;
        hence ex a, b being Element of Omega S st a = Fai.j & b = Fbi.j & a <=
        b by A22,A21,A23;
      end;
      then Fai <= Fbi;
      hence (F.a).i <= (F.b).i by A20,Th3;
    end;
    hence thesis by WAYBEL_3:28;
  end;
A24: the carrier of XxS9xM = the carrier of XxS|^M by YELLOW_1:def 5
    .= Funcs(M, the carrier of XxS) by YELLOW_1:28;
  then
A25: the carrier of XxS9xM c= Funcs(M, Funcs(the carrier of X, the carrier
  of S)) by Th32,FUNCT_5:56;
A26: G is monotone
  proof
    let a,b be Element of XxS9xM such that
A27: a <= b;
    reconsider f = G.a, g = G.b as continuous Function of X, Omega S9M by Th1;
    now
      let i be set;
      assume i in the carrier of X;
      then reconsider x = i as Element of X;
      reconsider fx = f.x, gx = g.x as Element of M-POS_prod(M --> Omega S) by
A17;
      now
        let j be Element of M;
A28:    (M --> XxS).j = XxS by FUNCOP_1:7;
        then reconsider
        aj = a.j, bj = b.j as continuous Function of X, Omega S by Th1;
        a.j <= b.j by A27,WAYBEL_3:28;
        then aj <= bj by A28,Th3;
        then
A29:    ex a, b being Element of Omega S st a = aj.x & b = bj.x & a <= b;
        b in the carrier of XxS9xM & G.b = commute b by A5;
        then
A30:    gx.j = bj.x by A25,FUNCT_6:56;
        a in the carrier of XxS9xM & G.a = commute a by A5;
        then fx.j = aj.x by A25,FUNCT_6:56;
        hence fx.j <= gx.j by A30,A29,FUNCOP_1:7;
      end;
      then fx <= gx by WAYBEL_3:28;
      hence ex a,b being Element of Omega S9M st a = f.i & b = g.i & a <= b by
A17,YELLOW_0:1;
    end;
    then f <= g;
    hence thesis by Th3;
  end;
  now
    let a be Element of XxS9xM;
    a in Funcs(M, the carrier of XxS) & Funcs(M, the carrier of XxS) c=
    Funcs(M, Funcs(the carrier of X, the carrier of S)) by A24,Th32,FUNCT_5:56;
    then
A31: commute commute a = a by FUNCT_6:57;
    thus (F*G).a = F.(G.a) by FUNCT_2:15
      .= commute (G.a) by A1
      .= a by A5,A31
      .= (id XxS9xM).a;
  end;
  then F*G = id XxS9xM by FUNCT_2:63;
  hence F is isomorphic by A18,A26,A16,YELLOW16:15;
  let f be continuous Function of X, M-TOP_prod (M --> Sierpinski_Space);
  f is Element of XxxS9M by Th2;
  hence thesis by A1;
end;
