reserve i, j, m, n, k for Nat,
  x, y for set,
  K for Field,
  a,a1 for Element of K;
reserve V1,V2,V3 for finite-dimensional VectSp of K,
  f for Function of V1,V2,

  b1,b19 for OrdBasis of V1,
  B1 for FinSequence of V1,
  b2 for OrdBasis of V2,
  B2 for FinSequence of V2,

  B3 for FinSequence of V3,
  v1,w1 for Element of V1,
  R,R1,R2 for FinSequence of V1,
  p,p1,p2 for FinSequence of K;

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;
