theorem Th57:
  u + v in v + W iff u in W
proof
  thus u + v in v + W implies u in W
  proof
    assume u + v in v + W;
    then ex v1 st u + v = v + v1 & v1 in W;
    hence thesis by RLVECT_1:8;
  end;
  assume u in W;
  hence thesis;
end;
