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
  for i, j st i in Seg n & j in Seg n & i <> j holds Det A = Det RLine(A
  ,i,Line(A,i) + a*Line(A,j))
proof
  let i, j such that
A1: i in Seg n and
A2: j in Seg n and
A3: i <> j;
A4: width A = n by MATRIX_0:24;
  then
A5: len Line(A,j)=n by MATRIX_0:def 7;
A6: len Line(A,j)=len (a*Line(A,j)) by Lm5;
  len Line(A,i)=n by A4,MATRIX_0:def 7;
  hence Det RLine(A,i,Line(A,i)+a*Line(A,j))= Det(RLine(A,i,Line(A,i))) + Det(
  RLine(A,i,a*Line(A,j))) by A1,A5,A6,Th36
    .=Det(A)+Det(RLine(A,i,a*Line(A,j))) by Th30
    .=Det(A)+0.K by A1,A2,A3,Th52
    .=Det(A) by RLVECT_1:4;
end;
