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

theorem Th39:
  for R,S being RMembership_Func of X,X, n being Nat st
  R c= S holds n iter R c= n iter S
proof
  let R,S be RMembership_Func of X,X;
  let n be Nat;
  defpred P[Nat] means $1 iter R c= $1 iter S;
  assume
A1: R c= S;
A2: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume
A3: P[k];
    (k iter R) (#) R = (k+1) iter R & (k iter S) (#) S = (k+1) iter S by Th26;
    hence thesis by A1,A3,Th6;
  end;
  0 iter R = Imf(X,X) by Th24
    .= 0 iter S by Th24;
  then
A4: P[0];
  for k being Nat holds P[k] from NAT_1:sch 2(A4,A2);
  hence thesis;
end;
