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;
