reserve x,y for set,
  i,j,k,l,m,n for Nat,
  K for Field,
  N for without_zero finite Subset of NAT,
  a,b for Element of K,
  A,B,B1,B2,X,X1,X2 for (Matrix of K),
  A9 for (Matrix of m,n,K),
  B9 for (Matrix of m,k,K);
reserve D for non empty set,
  bD for FinSequence of D,
  b,f,g for FinSequence of K,
  MD for Matrix of D;

theorem Th32:
  ColVec2Mx (k|->0.K) = 0.(K,k,1)
proof
  card(k|->0.K) = k by CARD_1:def 7;
  then reconsider C=ColVec2Mx (k|->0.K) as Matrix of k,1,K;
  set Z=0.(K,k,1);
  now
A1: len (k|->0.K)=k by CARD_1:def 7;
    let i,j such that
A2: [i,j] in Indices C;
A3: i in dom C by A2,ZFMISC_1:87;
A4: j in Seg width C by A2,ZFMISC_1:87;
A5: dom C=Seg len C & len C=len (k|->0.K) by FINSEQ_1:def 3,MATRIX_0:def 2;
    then width C=1 by A3,A1,Th26;
    then
A6: j=1 by A4,FINSEQ_1:2,TARSKI:def 1;
    Indices Z=Indices C by MATRIX_0:26;
    hence Z*(i,j) = 0.K by A2,MATRIX_3:1
      .= (k|->0.K).i by A3,A5,A1,FINSEQ_2:57
      .= Col(C,j).i by A3,A5,A1,A6,Th26
      .= C*(i,j) by A3,MATRIX_0:def 8;
  end;
  hence thesis by MATRIX_0:27;
end;
