reserve
  j, k, l, n, m, t,i for Nat,
  K for comRing, 
  a for Element of K,
  M,M1,M2 for Matrix of n,m,K,
  pK,qK for FinSequence of K,
  A for Matrix of n,K;

theorem
  l in dom (1.(K,n)) & n>0 implies A * SXCol((1.(K,n)),l,a) = SXCol(A,l, a)
proof
  assume that
A1: l in dom (1.(K,n)) and
A2: n>0;
A3: len (1.(K,n)) = n by MATRIX_0:24;
A4: width (1.(K,n)) = n by MATRIX_0:24;
  then
A5: dom (1.(K,n)) = Seg width (1.(K,n)) by A3,FINSEQ_1:def 3;
A6: width (A@)=n by MATRIX_0:24;
A7: width (SXLine((1.(K,n)),l,a)) = width (1.(K,n)) & len (A@)=n by Th1,
MATRIX_0:24;
A8: len A=n by MATRIX_0:24;
A9: width A=n by MATRIX_0:24;
  then
A10: Seg width A = dom (1.(K,n)) by A3,FINSEQ_1:def 3;
  (SXLine(A@,l,a))@ = (SXLine((1.(K,n)),l,a) * (A@))@ by A1,Th7
    .= (A@)@ * (SXLine((1.(K,n)),l,a))@ by A2,A4,A7,A6,MATRIX_3:22
    .= A* (SXLine((1.(K,n)),l,a))@ by A2,A8,A9,MATRIX_0:57
    .= A* (SXLine((1.(K,n))@,l,a))@ by MATRIX_6:10
    .= A* SXCol((1.(K,n)),l,a) by A1,A2,A5,Th16;
  hence thesis by A1,A2,A10,Th16;
end;
