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

theorem Th37:
  for R being RMembership_Func of X,X, n being Nat st R
  is transitive & n > 0 holds n iter R c= R
proof
  let R be RMembership_Func of X,X;
  let n be Nat;
  assume that
A1: R is transitive and
A2: n > 0;
  reconsider n as non zero Element of NAT by A2,ORDINAL1:def 12;
  defpred P[Nat] means $1 iter R c= R;
A3: R (#) R c= R by A1;
A4: for k being non zero Nat st P[k] holds P[k+1]
  proof
    let k be non zero Nat;
    assume P[k];
    then k iter R (#) R c= R(#)R by Th6;
    then (k+1) iter R c= R(#)R by Th26;
    hence thesis by A3,Th5;
  end;
A5: P[1] by Th25;
  for k being non zero Nat holds P[k] from NAT_1:sch 10(A5,A4);
  then P[n];
  hence thesis;
end;
