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 Th29:
  i in Seg n implies Line(M,j) "*" Col((Matrix_of_Cofactor M)@,i)
  = Det RLine(M,i,Line(M,j))
proof
  assume
A1: i in Seg n;
  set C=Matrix_of_Cofactor M;
  len C=n by MATRIX_0:24;
  then dom C=Seg n by FINSEQ_1:def 3;
  then
A2: Line(C,i)=Col(C@,i) by A1,MATRIX_0:58;
  width M=n by MATRIX_0:24;
  then
A3: len Line(M,j)=n by MATRIX_0:def 7;
  thus Det RLine(M,i,Line(M,j)) = Sum LaplaceExpL(RLine(M,i,Line(M,j)),i) by A1
,Th25
    .= Sum mlt(Col(C@,i),Line(M,j)) by A1,A2,A3,Th28
    .= Line(M,j) "*" Col(C@,i) by FVSUM_1:64;
end;
