reserve X, Y for non empty set;
reserve X for non empty set;
reserve R for RMembership_Func of X,X;

theorem Th35:
  for R being RMembership_Func of X,X, Q being Subset of
FuzzyLattice [:X,X:] holds @("\/"(Q,FuzzyLattice [:X,X:])) (#) R = "\/"({@r (#)
  R where r is Element of FuzzyLattice [:X,X:]:r in Q}, FuzzyLattice [:X,X:])
proof
  let R be RMembership_Func of X,X;
  let Q be Subset of FuzzyLattice [:X,X:];
  set FL = FuzzyLattice [:X,X:], RP = RealPoset [. 0,1 .];
  ("\/"({@r (#) R where r is Element of FL:r in Q},FL)) = @("\/"({@r (#) R
  where r is Element of FL:r in Q},FL)) by LFUZZY_0:def 5;
  then reconsider F = ("\/"({@r (#) R where r is Element of FL:r in Q},FL)) as
  RMembership_Func of X,X;
  for x,z being Element of X holds (@("\/"(Q,FL)) (#) R).(x,z) = F.(x,z)
  proof
    let x,z be Element of X;
A1: {(@r(#)R) where r is Element of FL: r in Q} c= the carrier of
    FuzzyLattice [:X,X:]
    proof
      let t be object;
      assume t in {(@r(#)R) where r is Element of FL: r in Q};
      then consider r being Element of FuzzyLattice [:X,X:] such that
A2:   t = (@r(#)R) and
      r in Q;
      ([:X,X:],(@r(#)R))@ = (@r(#)R) by LFUZZY_0:def 6;
      hence thesis by A2;
    end;
A3: the set of all "\/"(pi(Q, [x,y]), RP) "/\" R. [y,z] where y is Element of X
 = the set of all "\/"({b "/\" R. [y9,z] where b is Element of RP:b in pi(Q,[x,
    y9])} ,RP) where y9 is Element of X
    proof
      deffunc G(Element of X) = "\/"({b "/\" R. [$1,z] where b is Element of
RP:   b in pi(Q,[x,$1])} ,RP);
      deffunc F(Element of X) = "\/"(pi(Q, [x,$1]),RP) "/\" R. [$1,z];
      defpred P[Element of X] means not contradiction;
      for y being Element of X holds "\/"(pi(Q, [x,y]), RealPoset [. 0,1
.]) "/\" R. [y,z] = "\/"({b "/\" R. [y,z] where b is Element of RP:b in pi(Q,[x
      ,y])} ,RP)
      proof
        FuzzyLattice [:X,X:] = (RealPoset [. 0,1 .]) |^ [:X,X:] by
LFUZZY_0:def 4
          .= product ([:X,X:] --> RealPoset [. 0,1 .]) by YELLOW_1:def 5;
        then reconsider
        Q as Subset of product ([:X,X:] --> RealPoset [. 0,1 .]);
        let y be Element of X;
        pi(Q, [x,y]) is Subset of RealPoset [. 0,1 .] by FUNCOP_1:7;
        hence thesis by LFUZZY_0:15;
      end;
      then
A4:   for y being Element of X holds F(y) = G (y);
      {F(y) where y is Element of X:P[y]} = {G(y9) where y9 is Element of
X:    P[y9]} from FRAENKEL:sch 5(A4);
      hence thesis;
    end;
A5: the set of all
"\/"({b "/\" R. [y,z] where b is Element of RP:b in pi(Q,[x,y])} ,RP
) where y is Element of X = the set of all "\/"({r. [x,y9] "/\" R. [y9,z]
    where r is Element of FL: r in Q} ,RP) where y9 is Element of X
    proof
      deffunc G(Element of X) = "\/"({r. [x,$1] "/\" R. [$1,z] where r is
      Element of FL: r in Q} ,RP);
      deffunc F(Element of X) = "\/"({b "/\" R. [$1,z] where b is Element of
RP:   b in pi(Q,[x,$1])} ,RP);
      defpred P[Element of X] means not contradiction;
      for y being Element of X holds "\/"({b "/\" R. [y,z] where b is
Element of RP: b in pi(Q,[x,y])} ,RP) = "\/"({r. [x,y] "/\" R. [y,z] where r is
      Element of FL: r in Q} ,RP)
      proof
        let y be Element of X;
        set A = {b "/\" R. [y,z] where b is Element of RP: b in pi(Q,[x,y])},
        B = {r. [x,y] "/\" R. [y,z] where r is Element of FL: r in Q};
A6:     B c= A
        proof
          let a be object;
          assume a in B;
          then consider r being Element of FL such that
A7:       a = r. [x,y] "/\" R. [y,z] and
A8:       r in Q;
          r. [x,y] in pi(Q,[x,y]) by A8,CARD_3:def 6;
          hence thesis by A7;
        end;
        A c= B
        proof
          let a be object;
          assume a in A;
          then consider b be Element of RP such that
A9:       a = b "/\" R. [y,z] and
A10:      b in pi(Q,[x,y]);
          ex f be Function st f in Q & b = f. [x,y] by A10,CARD_3:def 6;
          hence thesis by A9;
        end;
        hence thesis by A6,XBOOLE_0:def 10;
      end;
      then
A11:  for y being Element of X holds F(y) = G (y);
      thus {F(y) where y is Element of X:P[y]} = {G(y) where y is Element of X
      :P[y]} from FRAENKEL:sch 5(A11);
    end;
A12: "\/"((the set of all "\/"({r. [x,y] "/\" R. [y,z]
     where r is Element of FL: r in Q}
    ,RP) where y is Element of X),RP) = "\/"({"\/"((the set of all r9. [x,y]
    "/\" R. [y,z] where y is Element of X),RP) where r9 is
    Element of FL: r9 in Q},RP)
    proof
      deffunc F(Element of X,Element of FuzzyLattice [:X,X:]) = $2. [x,$1]
      "/\" R. [$1,z];
      defpred Q[Element of FuzzyLattice [:X,X:]] means $1 in Q;
      defpred P[Element of X] means not contradiction;
A13:  for y being Element of X, r being Element of FL st P[y] & Q[r]
      holds F(y,r) = F(y,r);
      thus "\/"({ "\/"({F(y,r) where r is Element of FL: Q[r]}, RP) where y
is Element of X: P[y] }, RP) = "\/"({ "\/"({F(y9,r9) where y9 is Element of X:
P[y9]}, RP) where r9 is Element of FL: Q[r9] }, RP) from LFUZZY_0:sch 5(A13);
    end;
A14: {"\/"((the set of all r. [x,y] "/\" R. [y,z] where y is Element of X),RP)
     where r is Element of FL: r in Q} = {"\/"((the set of all @r9. [x,y] "/\"
R. [y,z] where y is Element of X) ,RP) where r9 is Element of
FL: r9 in Q}
    proof
      deffunc G(Element of FL) = "\/"((the set of all
      @($1). [x,y] "/\" R. [y,z] where y is
      Element of X),RealPoset [. 0,1 .]);
      deffunc F(Element of FL) = "\/"((the set of all
      $1. [x,y] "/\" R. [y,z] where y is
      Element of X),RealPoset [. 0,1 .]);
      defpred P[Element of FL] means $1 in Q;
      for r being Element of FL st r in Q holds "\/"((the set of all
      r. [x,y] "/\" R. [y
,z] where y is Element of X) ,RealPoset [. 0,1 .]) = "\/"((the set of all @r
. [x,y] "/\" R. [y,z] where y is Element of X) ,RealPoset [.
      0,1 .])
      proof
        let r be Element of FL;
        assume r in Q;
        the set of all r. [x,y] "/\" R. [y,z] where y is Element of X =the set
of all @r. [x,y] "/\" R. [y,z] where y is Element of X
        proof
          deffunc g(Element of X) = @r. [x,$1] "/\" R. [$1,z];
          deffunc f(Element of X) = r. [x,$1] "/\" R. [$1,z];
          defpred P[Element of X] means not contradiction;
A15:      for y being Element of X holds f(y) = g(y) by LFUZZY_0:def 5;
          thus {f(y) where y is Element of X:P[y]} ={g(y) where y is Element
          of X:P[y]} from FRAENKEL:sch 5(A15);
        end;
        hence thesis;
      end;
      then
A16:  for r being Element of FL st P[r] holds F(r) = G(r);
      thus {F(r) where r is Element of FuzzyLattice [:X,X:]: P[r] } = {G(r)
where r is Element of FuzzyLattice [:X,X:]: P[r]} from FRAENKEL:sch 6(A16);
    end;
A17: {(@r(#)R). [x,z] where r is Element of FL: r in Q} = pi({(@r(#)R)
    where r is Element of FL: r in Q}, [x,z])
    proof
      set A= {(@r(#)R). [x,z] where r is Element of FL: r in Q}, B= pi({(@r(#)
      R) where r is Element of FL: r in Q}, [x,z]);
      thus A c= B
      proof
        let a be object;
        assume a in A;
        then consider r being Element of FL such that
A18:    a = (@r(#)R). [x,z] and
A19:    r in Q;
        (@r(#)R) in {(@r9(#)R) where r9 is Element of FL: r9 in Q} by A19;
        hence thesis by A18,CARD_3:def 6;
      end;
      thus B c= A
      proof
        let b be object;
        assume b in B;
        then consider f be Function such that
A20:    f in {(@r9(#)R) where r9 is Element of FL: r9 in Q} and
A21:    b = f. [x,z] by CARD_3:def 6;
        ex r being Element of FL st f = (@r(#)R) & r in Q by A20;
        hence thesis by A21;
      end;
    end;
A22: the set of all
(@("\/"(Q,FL))). [x,y] "/\" R. [y,z] where y is Element of X = the set of all
"\/"(pi(Q, [x,y]), RP) "/\" R. [y,z] where y is Element of X
    proof
      deffunc G(Element of X) = "\/"(pi(Q, [x,$1]), RP) "/\" R. [$1,z];
      deffunc F(Element of X) = (@("\/"(Q,FL))). [x,$1] "/\" R. [$1,z];
      defpred P[Element of X] means not contradiction;
      for y being Element of X holds (@("\/"(Q,FuzzyLattice [:X,X:]))). [x
      ,y] "/\" R. [y,z] = "\/"(pi(Q, [x,y]), RealPoset [. 0,1 .]) "/\" R. [y,z]
      proof
        let y be Element of X;
        (@("\/"(Q,FuzzyLattice [:X,X:]))). [x,y] = (("\/"(Q,FuzzyLattice
        [:X,X:]))). [x,y] by LFUZZY_0:def 5
          .= "\/"(pi(Q, [x,y]), RealPoset [. 0,1 .]) by Th32;
        hence thesis;
      end;
      then
A23:  for y being Element of X holds F(y) = G(y);
      thus {F(y) where y is Element of X: P[y]} = {G(y9) where y9 is Element
      of X: P[y9]} from FRAENKEL:sch 5 (A23);
    end;
A24: {"\/"((the set of all @r. [x,y] "/\" R. [y,z] where y is Element of X),RP)
     where r is Element of FL: r in Q} = {(@r9(#)R). [x,z] where
    r9 is Element of FL: r9 in Q}
    proof
      deffunc G(Element of FL) = (@($1)(#)R). [x,z];
      deffunc F(Element of FL) = "\/"((the set of all @($1). [x,y] "/\"
      R. [y,z] where y is Element of X),RealPoset [. 0,1 .]);
      defpred P[Element of FL] means $1 in Q;
A25:  for r being Element of FL st P[r] holds F(r) = G(r) by Lm6;
      thus {F(r) where r is Element of FuzzyLattice [:X,X:]:P[r]} = {G(r)
where r is Element of FuzzyLattice [:X,X:]: P[r]} from FRAENKEL:sch 6(A25);
    end;
    thus (@("\/"(Q,FL)) (#) R).(x,z) = "\/"((the set of all
    (@("\/"(Q,FL))). [x,y] "/\" R. [y
    ,z] where y is Element of X),RP) by Lm6
      .= F.(x,z) by A1,A22,A3,A5,A12,A14,A24,A17,Th32;
  end;
  hence thesis by Th2;
end;
