reserve S for 1-sorted,
  i for Element of NAT,
  p for FinSequence,
  X for set;

theorem Th32:
  for X being set, u,v being Element of bspace(X), x being Element
  of X holds (u + v)@x = u@x + v@x
proof
  let X be set, u,v be Element of bspace(X), x be Element of X;
  reconsider u9 = u, v9 = v as Subset of X;
  (u + v)@x = (u9 \+\ v9)@x by Def5
    .= (u9@x) + (v9@x) by Th15;
  hence thesis;
end;
