theorem Th78:
  v in W implies z * v in v + W
proof
  assume v in W;
  then
A1: (z - 1r) * v in W by Th40;
  z * v = ((z - 1r) + 1r) * v .= (z - 1r) * v + 1r * v by Def3
    .= v + (z - 1r) * v by Def5;
  hence thesis by A1;
end;
