
theorem
  for M being non empty set, T being non empty TopSpace holds the RelStr
  of Omega product(M --> T) = the RelStr of product(M --> Omega T)
proof
  let M be non empty set, T be non empty TopSpace;
A1: dom Carrier (M --> T) = M by PARTFUN1:def 2
    .= dom Carrier (M --> Omega T) by PARTFUN1:def 2;
A2: for i being object
  st i in dom Carrier (M --> T) holds (Carrier (M --> T)).
  i = (Carrier (M --> Omega T)).i
  proof
    let i be object;
    assume i in dom Carrier (M --> T);
    then
A3: i in M;
    then consider R1 being 1-sorted such that
A4: R1 = (M --> T).i and
A5: (Carrier (M --> T)).i = the carrier of R1 by PRALG_1:def 15;
    consider R2 being 1-sorted such that
A6: R2 = (M --> Omega T).i and
A7: (Carrier (M --> Omega T)).i = the carrier of R2 by A3,PRALG_1:def 15;
    the carrier of R1 = the carrier of T by A3,A4,FUNCOP_1:7
      .= the carrier of Omega T by Lm1
      .= the carrier of R2 by A3,A6,FUNCOP_1:7;
    hence thesis by A5,A7;
  end;
A8: the carrier of Omega product (M --> T) = the carrier of product (M -->
  T) by Lm1
    .= product Carrier (M --> T) by WAYBEL18:def 3
    .= product Carrier (M --> Omega T) by A1,A2,FUNCT_1:2;
A9: the carrier of Omega product (M --> T) = the carrier of product (M -->
  T) by Lm1;
  the InternalRel of Omega product(M --> T) = the InternalRel of product
  (M --> Omega T)
  proof
    let x, y be object;
    hereby
      assume
A10:  [x,y] in the InternalRel of Omega product(M --> T);
      then
A11:  y in the carrier of Omega product(M --> T) by ZFMISC_1:87;
A12:  x in the carrier of Omega product(M --> T) by A10,ZFMISC_1:87;
      then reconsider
      xp = x, yp = y as Element of product(M --> Omega T) by A8,A11,
YELLOW_1:def 4;
      reconsider p1 = x, p2 = y as Element of product(M --> T) by A12,A11,Lm1;
      reconsider xo = x, yo = y as Element of Omega product(M --> T) by A10,
ZFMISC_1:87;
A13:  xp in product Carrier (M --> Omega T) by A8,A10,ZFMISC_1:87;
      then consider f being Function such that
A14:  xp = f and
      dom f = dom Carrier (M --> Omega T) and
      for i being object st i in dom Carrier (M --> Omega T) holds f.i in
      Carrier (M --> Omega T).i by CARD_3:def 5;
      yp in product Carrier (M --> Omega T) by A8,A10,ZFMISC_1:87;
      then consider g being Function such that
A15:  yp = g and
      dom g = dom Carrier (M --> Omega T) and
      for i being object st i in dom Carrier (M --> Omega T) holds g.i in
      Carrier (M --> Omega T).i by CARD_3:def 5;
      xo <= yo by A10;
      then
A16:  ex Yp being Subset of product(M --> T) st Yp = {p2} & p1 in Cl Yp
      by Def2;
      for i being object st i in M ex R being RelStr, xi, yi being Element
      of R st R = (M --> Omega T).i & xi = f.i & yi = g.i & xi <= yi
      proof
        let i be object;
        assume
A17:    i in M;
        then reconsider j = i as Element of M;
        set t1 = p1.j, t2 = p2.j;
        reconsider xi = t1, yi = t2 as Element of Omega T by Lm1;
        take Omega T, xi, yi;
        thus Omega T = (M --> Omega T).i by A17,FUNCOP_1:7;
        thus xi = f.i by A14;
        thus yi = g.i by A15;
        p1.j in Cl {p2.j} by A16,YELLOW14:30;
        hence xi <= yi by Def2;
      end;
      then xp <= yp by A13,A14,A15,YELLOW_1:def 4;
      hence [x,y] in the InternalRel of product (M --> Omega T);
    end;
    assume
A18: [x,y] in the InternalRel of product (M --> Omega T);
    then reconsider xp = x, yp = y as Element of product(M --> Omega T) by
ZFMISC_1:87;
A19: xp <= yp by A18;
A20: x in the carrier of product (M --> Omega T) by A18,ZFMISC_1:87;
    then xp in product Carrier (M --> Omega T) by YELLOW_1:def 4;
    then consider f, g being Function such that
A21: f = xp and
A22: g = yp and
A23: for i being object st i in M ex R being RelStr, xi, yi being Element
    of R st R = (M --> Omega T).i & xi = f.i & yi = g.i & xi <= yi by A19,
YELLOW_1:def 4;
A24: y in the carrier of product (M --> Omega T) by A18,ZFMISC_1:87;
    then reconsider
    xo = x, yo = y as Element of Omega product(M --> T) by A8,A20,
YELLOW_1:def 4;
    reconsider p1 = x, p2 = y as Element of product(M --> T) by A8,A9,A20,A24,
YELLOW_1:def 4;
    for i being Element of M holds p1.i in Cl {p2.i}
    proof
      let i be Element of M;
      consider R1 being RelStr, xi, yi being Element of R1 such that
A25:  R1 = (M --> Omega T).i and
A26:  xi = f.i and
A27:  yi = g.i and
A28:  xi <= yi by A23;
      reconsider xi, yi as Element of Omega T by A25;
      xi <= yi by A25,A28;
      then ex Y being Subset of T st Y = {yi} & xi in Cl Y by Def2;
      hence thesis by A21,A22,A26,A27;
    end;
    then p1 in Cl {p2} by YELLOW14:30;
    then xo <= yo by Def2;
    hence thesis;
  end;
  hence thesis by A8,YELLOW_1:def 4;
end;
