theorem Th50:
  (u1 + v1) + w1 = u1 + (v1 + w1)
proof
  reconsider u = u1, v = v1, w = w1 as Vector of M by Th48;
A1: v1 + w1 = v + w by Def13;
  u1 + v1 = u + v by Def13;
  hence (u1 + v1) + w1 = (u + v) + w by Def13
    .= u + (v + w) by Th43
    .= u1 + (v1 + w1) by A1,Def13;
end;
