reserve k,m,n for Nat,
  a, b, c for object,
  x, y, X, Y, Z for set,
  D for non empty set;
reserve p, q, r, s, t, u, v for FinSequence;
reserve P, Q, R, P1, P2, Q1, Q2, R1, R2 for FinSequence-membered set;
reserve S, T for non empty FinSequence-membered set;

theorem Th10:
  for P, m, n holds P^^(m+n) = (P^^m)^(P^^n)
proof
  let P, m;
  defpred X[ Nat ] means P^^(m+$1) = (P^^m)^(P^^$1);
  A1: X[ 0 ]
    proof
    thus P^^(m+0) = (P^^m)^{{}} by Th3 .= (P^^m)^(P^^0) by Th6;
    end;
  A20: for k holds X[ k ] implies X[ k+1 ]
    proof
    let k;
    assume A21: P^^(m+k) = (P^^m)^(P^^k);
    thus P^^(m+(k+1)) = P^^((m+k)+1)
        .= ((P^^m)^(P^^k))^P by Th6, A21
        .= (P^^m)^((P^^k)^P) by Th2
        .= (P^^m)^(P^^(k+1)) by Th6;
    end;
  for k holds X[ k ] from NAT_1:sch 2(A1, A20);
  hence thesis;
end;
