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

theorem Th38:
  for R being RMembership_Func of X,X st R is transitive holds R = TrCl R
proof
  let R be RMembership_Func of X,X;
  assume
A1: R is transitive;
  for c being Element of [:X,X:] holds (TrCl R).c <= R.c
  proof
    set Q = {n iter R where n is Element of NAT : n > 0}, RP = RealPoset [. 0,
    1 .];
    let c be Element of [:X,X:];
    for b being Element of RP st b in pi(Q,c) holds b <<= R.c
    proof
      let b be Element of RP;
      assume b in pi(Q,c);
      then consider f being Function such that
A2:   f in Q and
A3:   b = f.c by CARD_3:def 6;
      consider n be Element of NAT such that
A4:   f = n iter R and
A5:   n>0 by A2;
      n iter R c= R by A1,A5,Th37;
      then (n iter R).c <= R.c;
      hence thesis by A3,A4,LFUZZY_0:3;
    end;
    then
A6: R.c is_>=_than pi(Q,c) by LATTICE3:def 9;
    Q c= the carrier of FuzzyLattice [:X,X:]
    proof
      let t be object;
      assume t in Q;
      then consider i being Element of NAT such that
A7:   t = (i iter R) and
      i > 0;
      ([:X,X:],(i iter R))@ = (i iter R) by LFUZZY_0:def 6;
      hence thesis by A7;
    end;
    then (TrCl R).c = "\/"(pi(Q,c),RP) by Th32;
    then (TrCl R).c <<= R.c by A6,YELLOW_0:32;
    hence thesis by LFUZZY_0:3;
  end;
  then
A8: TrCl R c= R;
  R c= TrCl R by Th30;
  hence thesis by A8,Th3;
end;
