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

theorem Th29:
  for x,y being Element of X holds (TrCl R). [x,y] = "\/"(pi({n
  iter R where n is Element of NAT : n > 0}, [x,y]), RealPoset [. 0,1 .])
proof
  set Q = {n iter R where n is Element of NAT : n > 0};
  Q c= the carrier of FuzzyLattice [:X,X:]
  proof
    let t be object;
    assume t in Q;
    then ex n being Element of NAT st t = n iter R & n > 0;
    then reconsider t as Membership_Func of [:X,X:];
    ([:X,X:],t)@ is Element of FuzzyLattice [:X,X:];
    then reconsider t as Element of FuzzyLattice [:X,X:] by LFUZZY_0:def 6;
    t in the carrier of FuzzyLattice [:X,X:];
    hence thesis;
  end;
  then reconsider Q as Subset of FuzzyLattice [:X,X:];
  let x,z be Element of X;
  reconsider i = [x,z] as Element of [:X,X:];
A1: for a being Element of [:X,X:] holds ([:X,X:] --> RealPoset [. 0,1 .]).a
  is complete LATTICE by FUNCOP_1:7;
  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 (sup Q).i = "\/"(pi(Q,i), ([:X,X:] --> RealPoset [. 0,1 .]).i) by A1,
WAYBEL_3:32;
  hence thesis by FUNCOP_1:7;
end;
