theorem Th81:
  for A being 0-based finite array holds last (A^<%x%>) = x
  proof
    let A be 0-based finite array;
    dom <%x%> = 1 by AFINSQ_1:def 4; then
    dom(A^<%x%>) = (dom A)+^1 by ORDINAL4:def 1
    .= succ dom A by ORDINAL2:31; then
    union dom (A^<%x%>) = len A by ORDINAL2:2;
    hence thesis by AFINSQ_1:36;
  end;
