reserve x, y for object, X for set,
  i, j, k, l, n, m for Nat,
  D for non empty set,
  K for commutative Ring,
  a,b for Element of K,
  perm, p, q for Element of Permutations(n),
  Perm,P for Permutation of Seg n,
  F for Function of Seg n,Seg n,
  perm2, p2, q2, pq2 for Element of Permutations(n+2),
  Perm2 for Permutation of Seg (n+2);
reserve s for Element of 2Set Seg (n+2);
reserve pD for FinSequence of D,
  M for Matrix of n,m,D,
  pK,qK for FinSequence of K,
  A for Matrix of n,K;

theorem
  l in Seg n implies Det(RLine(A,l,a*Line(A,l))) = a*Det(A)
proof
A1: len Line(A,l)=width A by MATRIX_0:def 7;
  assume l in Seg n;
  then Det(RLine(A,l,a*Line(A,l)))=a*Det(RLine(A,l,Line(A,l))) by A1,Th34,
MATRIX_0:24;
  hence thesis by Th30;
end;
