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

theorem Th27:
  for m,n being Nat holds (m+n) iter R = (m iter R) (#) (n iter R)
proof
  let m,n be Nat;
  defpred P[Nat] means (m+ $1) iter R = (m iter R) (#) ($1 iter R);
A1: for n being Nat st P[n] holds P[n+1]
  proof
    let n be Nat;
    assume
A2: (m+n) iter R = (m iter R) (#) (n iter R);
    thus ((m) iter R) (#) ((n+1) iter R) = (m iter R) (#) ((n iter R) (#)R )
    by Th26
      .= ((m+n) iter R) (#) R by A2,LFUZZY_0:23
      .= ((m+n)+1) iter R by Th26
      .= (m+(n+1)) iter R;
  end;
  (m iter R) (#) (0 iter R) = (m iter R) (#) Imf(X,X) by Th24
    .= m iter R by Th23;
  then
A3: P[0];
  for m being Nat holds P[m] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
