theorem Th59:
  v in W implies a * v in v + W
proof
  assume v in W;
  then
A1: (a - 1) * v in W by Th21;
  a * v = ((a - 1) + 1) * v .= (a - 1) * v + 1 * v by RLVECT_1:def 6
    .= v + (a - 1) * v by RLVECT_1:def 8;
  hence thesis by A1;
end;
