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)) & k in dom (1.(K,n)) & n>0 implies (ICol((1.(K,n)),
  l,k))~ = ICol((1.(K,n)),l,k)
proof
  assume that
A1: l in dom (1.(K,n)) & k in dom (1.(K,n)) and
A2: n>0;
A3: (ILine((1.(K,n)),l,k))~ = ILine((1.(K,n)),l,k) by A1,Th12;
a2: ILine((1.(K,n)),l,k) is invertible by A1,Th12;
  len (1.(K,n)) = n & width (1.(K,n)) = n by MATRIX_0:24;
  then
A4: dom (1.(K,n)) = Seg width (1.(K,n)) by FINSEQ_1:def 3;
  (1.(K,n))@ = 1.(K,n) by MATRIX_6:10;
  then ICol((1.(K,n)),l,k) = (ILine((1.(K,n)),l,k))@ by A1,A2,A4,Th15
    .= (ILine((1.(K,n)),l,k)@)~ by A3,MATRIX_6:13,a2
    .= (ILine((1.(K,n))@,l,k)@)~ by MATRIX_6:10
    .= (ICol((1.(K,n)),l,k))~ by A1,A2,A4,Th15;
  hence thesis;
end;
