reserve x,y for object,
  N for Element of NAT,
  c,i,j,k,m,n for Nat,
  D for non empty set,
  s for Element of 2Set Seg (n+2),
  p for Element of Permutations(n) ,
  p1, q1 for Element of Permutations(n+1),
  p2 for Element of Permutations(n +2),
  K for Field,
  a for Element of K,
  f for FinSequence of K,
  A for (Matrix of K),
  AD for Matrix of n,m,D,
  pD for FinSequence of D,
  M for Matrix of n,K;

theorem
  for j be Nat, M be Matrix of n,K st j in Seg n holds Det M = Sum
  LaplaceExpC(M,j)
proof
  let j be Nat, M be Matrix of n,K such that
A1: j in Seg n;
  thus Det M = Det M@ by MATRIXR2:43
    .= Sum LaplaceExpL(M@,j) by A1,Th25
    .= Sum LaplaceExpC(M,j) by A1,Th26;
end;
