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 Th12:
  l in dom (1.(K,n)) & k in dom (1.(K,n)) implies ILine((1.(K,n)),
  l,k) is invertible & (ILine((1.(K,n)),l,k))~ = ILine((1.(K,n)),l,k)
proof
  assume l in dom (1.(K,n)) & k in dom (1.(K,n));
  then
  ILine((1.(K,n)),l,k) * ILine((1.(K,n)),l,k) = ILine(ILine((1.(K,n)),l,k)
  ,l,k ) & ILine(ILine((1.(K,n)),l,k),l,k) = 1.(K,n) by Th6,Th11;
  then
A1: ILine((1.(K,n)),l,k) is_reverse_of ILine((1.(K,n)),l,k) by MATRIX_6:def 2;
  then ILine((1.(K,n)),l,k) is invertible by MATRIX_6:def 3;
  hence thesis by A1,MATRIX_6:def 4;
end;
