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

theorem
  for R,S being RMembership_Func of X,X st S is transitive & R c= S
  holds TrCl R c= S
proof
  let R,S be RMembership_Func of X,X;
  assume that
A1: S is transitive and
A2: R c= S;
  for c being Element of [:X,X:] holds (TrCl R).c <= (TrCl S).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 <<= (TrCl S).c
    proof
      let b be Element of RP;
      assume b in pi(Q,c);
      then consider f being Function such that
A3:   f in Q and
A4:   b = f.c by CARD_3:def 6;
      consider n be Element of NAT such that
A5:   f = n iter R and
A6:   n>0 by A3;
A7:   n iter S c= TrCl S by A6,Th31;
      n iter R c= n iter S by A2,Th39;
      then n iter R c= TrCl S by A7,Th5;
      then (n iter R).c <= (TrCl S).c;
      hence thesis by A4,A5,LFUZZY_0:3;
    end;
    then (TrCl S).c is_>=_than pi(Q,c) by LATTICE3:def 9;
    then
A8: "\/"(pi(Q,c),RP) <<= (TrCl S).c by YELLOW_0:32;
    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
A9:   t = (i iter R) and
      i > 0;
      ([:X,X:],(i iter R))@ = (i iter R) by LFUZZY_0:def 6;
      hence thesis by A9;
    end;
    then (TrCl R).c = "\/"(pi(Q,c),RP) by Th32;
    hence thesis by A8,LFUZZY_0:3;
  end;
  then TrCl R c= TrCl S;
  hence thesis by A1,Th38;
end;
