theorem Th42:
  V1 is trivial iff dim V1 = 0
proof
  hereby
    assume
A1: V1 is trivial;
    the carrier of V1 c= {0.V1}
    proof
      let x be object;
      assume
A2:   x in the carrier of V1;
      x=0.V1 by A1,A2;
      hence thesis by TARSKI:def 1;
    end;
    then the carrier of (Omega).V1 = {0.V1} by ZFMISC_1:33
      .= the carrier of (0).V1 by VECTSP_4:def 3;
    then (Omega).V1=(0).V1 by VECTSP_4:29;
    hence dim V1=0 by VECTSP_9:29;
  end;
  assume dim V1=0;
  then
A3: (Omega).V1=(0).V1 by VECTSP_9:29;
  for v1 holds v1=0.V1 by VECTSP_4:35,A3,STRUCT_0:def 5;
  hence thesis;
end;
