reserve i,j,x,y for object,
  f,g for Function;
reserve T,T1 for finite Tree,
  t,p for Element of T,
  t1 for Element of T1;

theorem
  for T, T1 being finite DecoratedTree, p being Element of dom T holds
  card(T with-replacement (p,T1)) + card (T|p) = card T + card T1
proof
  let T, T1 be finite DecoratedTree, p be Element of dom T;
A1: card dom T = card T & card dom T1 = card T1 by CARD_1:62;
A2: card dom (T with-replacement(p,T1)) = card(T with-replacement (p,T1)) &
  card dom (T|p) = card (T|p) by CARD_1:62;
  dom (T with-replacement(p, T1)) = dom T with-replacement (p, dom T1) &
  dom ( T|p) = (dom T)|p by TREES_2:def 10,def 11;
  hence thesis by A1,A2,Th16;
end;
