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

theorem Th25:
  for a being Element of Z_2, x,y being Element of bspace(X) holds
  a*(x+y) = (a*x)+(a*y)
proof
  let a be Element of Z_2, x,y be Element of bspace(X);
  reconsider c = x, d = y as Subset of X;
  a*(x+y) = a \*\ (c \+\ d) by Lm2
    .= (a \*\ c) \+\ (a \*\ d) by Th17
    .= (a*x)+(a*y) by Lm2;
  hence thesis;
end;
