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 Th34:
  for K being commutative Ring,
  pK being FinSequence of K,
  a being Element of K,
  A being Matrix of n, K holds
  l in Seg n & len pK = n implies Det(RLine(A,l,a*pK)) = a*Det( RLine(A,l,pK))
proof
  let K be commutative Ring,
  pK be FinSequence of K,
  a be Element of K,
  A be Matrix of n,K;
  assume that
A1: l in Seg n and
A2: len pK = n;
  pK is Element of (len pK)-tuples_on the carrier of K by FINSEQ_2:92;
  then
A3: a*pK+0.K*pK=(a+0.K)*pK by FVSUM_1:55;
  a+0.K=a by RLVECT_1:4;
  hence
  Det(RLine(A,l,a*pK))=a*Det(RLine(A,l,pK))+0.K*Det(RLine(A,l,pK)) by A1,A2,A3
,Th33
    .=a*Det(RLine(A,l,pK)) by RLVECT_1:4;
end;
