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 Th30:
  a * ColVec2Mx f = ColVec2Mx (a*f)
proof
A1: len f=len (a*f) by MATRIXR1:16;
  per cases;
  suppose
A2: len f=0;
    len (ColVec2Mx f)=len (a*ColVec2Mx f) by MATRIX_3:def 5;
    then
A3: a * ColVec2Mx f = {} by A2,MATRIX_0:def 2;
    len (ColVec2Mx (a*f))= 0 by A1,A2,MATRIX_0:def 2;
    hence thesis by A3;
  end;
  suppose
A4: len f>0;
A5: width (a*ColVec2Mx f)=width ColVec2Mx f by MATRIX_3:def 5;
A6: width ColVec2Mx f=1 by A4,MATRIX_0:23;
    then Col(a * ColVec2Mx f,1) = a*Col(ColVec2Mx f,1) by MATRIXR1:19
      .= a*f by A4,Th26;
    hence thesis by A1,A4,A6,A5,Th26;
  end;
end;
