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 Th36:
  l in Seg n & len pK = n & len qK = n implies Det(RLine(A,l,pK+qK
  )) = Det(RLine(A,l,pK)) + Det(RLine(A,l,qK))
proof
  assume that
A1: l in Seg n and
A2: len pK=n and
A3: len qK=n;
  pK is Element of (len pK)-tuples_on the carrier of K by FINSEQ_2:92;
  then
A4: 1_K*pK=pK by FVSUM_1:57;
  qK is Element of (len pK)-tuples_on the carrier of K by A2,A3,FINSEQ_2:92;
  then 1_K*qK=qK by FVSUM_1:57;
  hence Det(RLine(A,l,pK+qK))=1_K*Det(RLine(A,l,pK))+1_K*Det(RLine(A,l,qK)) by
A1,A2,A3,A4,Th33
    .= Det(RLine(A,l,pK))+1_K*Det(RLine(A,l,qK))
    .= Det(RLine(A,l,pK))+Det(RLine(A,l,qK));
end;
