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

theorem Th31:
  for n being Nat st n > 0 holds n iter R c= TrCl R
proof
  let n9 be Nat;
  assume
A1: n9 > 0;
  for c being Element of [: X,X :] holds (n9 iter R).c <= (TrCl R).c
  proof
    reconsider n9 as Element of NAT by ORDINAL1:def 12;
    set Q = {n iter R where n is Element of NAT : n > 0};
    let c be Element of [: X,X :];
    consider x,y being object such that
A2: [x,y] = c by RELAT_1:def 1;
    reconsider x,y as Element of X by A2,ZFMISC_1:87;
    n9 iter R in Q by A1;
    then
A3: (n9 iter R). [x,y] in pi(Q, [x,y]) by CARD_3:def 6;
    (TrCl R). [x,y] = "\/"(pi(Q, [x,y]), RealPoset [. 0,1 .]) by Th29;
    then (n9 iter R). [x,y] <<= (TrCl R). [x,y] by A3,YELLOW_2:22;
    hence thesis by A2,LFUZZY_0:3;
  end;
  hence thesis;
end;
