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)) & k<>l & n>0 implies (RColXS((
  1.(K,n)),l,k,a))~ = RColXS((1.(K,n)),l,k,-a)
proof
  assume that
A1: l in dom (1.(K,n)) & k in dom (1.(K,n)) and
A2: k<>l and
A3: n>0;
A4: (RLineXS((1.(K,n)),l,k,a))~ = RLineXS((1.(K,n)),l,k,-a) by A1,A2,Th13;
a3: RLineXS((1.(K,n)),l,k,a) is invertible by A1,Th13,A2;
  len (1.(K,n)) = n & width (1.(K,n)) = n by MATRIX_0:24;
  then
A5: 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 RColXS((1.(K,n)),l,k,-a) = (RLineXS((1.(K,n)),l,k,-a))@ by A1,A3,A5,Th17
    .= (RLineXS((1.(K,n)),l,k,a)@)~ by A4,MATRIX_6:13,a3
    .= (RLineXS((1.(K,n))@,l,k,a)@)~ by MATRIX_6:10
    .= (RColXS((1.(K,n)),l,k,a))~ by A1,A3,A5,Th17;
  hence thesis;
end;
