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

theorem Th32:
  for Q being Subset of FuzzyLattice X, x being Element of X holds
  ("\/"(Q,FuzzyLattice X)). x = "\/"(pi(Q, x), RealPoset [. 0,1 .])
proof
  let Q be Subset of FuzzyLattice X;
  let x be Element of X;
A1: for a being Element of X holds (X --> RealPoset [. 0,1 .]).a is complete
  LATTICE by FUNCOP_1:7;
  FuzzyLattice X = (RealPoset [. 0,1 .]) |^ X by LFUZZY_0:def 4
    .= product (X --> RealPoset [. 0,1 .]) by YELLOW_1:def 5;
  then (sup Q).x = "\/"(pi(Q,x), (X --> RealPoset [. 0,1 .]).x) by A1,
WAYBEL_3:32;
  hence thesis by FUNCOP_1:7;
end;
