theorem Th69:
  z <> 0 & (z * v) + W = the carrier of W implies v in W
proof
  assume that
A1: z <> 0 and
A2: (z * v) + W = the carrier of W;
  assume not v in W;
  then not 1r * v in W by Def5;
  then not (z" * z) * v in W by A1,XCMPLX_0:def 7;
  then not z" * (z * v) in W by Def4;
  then
A3: not z * v in W by Th40;
  0.V in W & z * v + 0.V = z * v by Th36,RLVECT_1:4;
  then z * v in {z * v + u : u in W};
  hence contradiction by A2,A3;
end;
