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

theorem Th34:
  for R being RMembership_Func of X,X, Q being Subset of
FuzzyLattice [:X,X:] holds R (#) @("\/"(Q,FuzzyLattice [:X,X:])) = "\/"({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 (R (#) @("\/"(Q,FL))).(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;
    thus (R (#) @("\/"(Q,FL))).(x,z) = "\/"((the set of all
R.(x,y) "/\" (@("\/"(Q,FL))) . (y
    ,z) where y is Element of X),RP) by LFUZZY_0:22
      .= "\/"((the set of all
R. [x,y] "/\" "\/"(pi(Q, [y,z]), RP) where y is Element of X),RP) by Lm2
      .= "\/"((the set of all
"\/"({R. [x,y] "/\" b where b is Element of RP:b in pi(Q,[y,
    z])} ,RP) where y is Element of X),RP) by Lm3
      .= "\/"((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) by Lm4
      .= "\/"( {"\/"((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},RP) by Lm10
      .= "\/"( {"\/"((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},RP) by Lm7
      .= "\/"({(R(#)@r). [x,z] where r is Element of FL: r in Q},RP) by Lm8
      .= "\/"(pi({(R(#)@r) where r is Element of FL: r in Q}, [x,z]),RP) by Lm9
      .= F.(x,z) by A1,Th32;
  end;
  hence thesis by Th2;
end;
