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 Th13:
  l in dom (1.(K,n)) & k in dom (1.(K,n)) & k<>l implies RLineXS((
1.(K,n)),l,k,a) is invertible & (RLineXS((1.(K,n)),l,k,a))~ = RLineXS((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;
  set B = RLineXS((1.(K,n)),l,k,-a);
  for i,j st [i,j] in Indices (1.(K,n)) holds (1.(K,n))*(i,j) = RLineXS(B,
  l,k,a)*(i,j) by A1,A2,Lm4;
  then
A3: 1.(K,n) = RLineXS(B,l,k,a) by MATRIX_0:27;
  set b = -a;
  set A = RLineXS((1.(K,n)),l,k,a);
  a+b = 0.K by RLVECT_1:def 10;
  then a = -b by RLVECT_1:6;
  then
  for i,j st [i,j] in Indices (1.(K,n)) holds (1.(K,n))*(i,j) = RLineXS(A,
  l,k,-a)*(i,j) by A1,A2,Lm4;
  then
A4: 1.(K,n) = RLineXS(A,l,k,-a) by MATRIX_0:27;
  A * B = RLineXS(B,l,k,a) & B * A = RLineXS(A,l,k,-a) by A1,Th8;
  then
A5: B is_reverse_of A by A3,A4,MATRIX_6:def 2;
  then A is invertible by MATRIX_6:def 3;
  hence thesis by A5,MATRIX_6:def 4;
end;
