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 Th32:
  i in Seg n & j in Seg n implies Line((Matrix_of_Cofactor M)@,i)
  "*" Col(M,j) = Det RLine(M@,i,Line(M@,j))
proof
  assume that
A1: i in Seg n and
A2: j in Seg n;
  set C=Matrix_of_Cofactor M;
  set L=Line(M@,j);
A3: width C = n by MATRIX_0:24;
  width (M@)=n by MATRIX_0:24;
  then
A4: len L=n by MATRIX_0:def 7;
A5: width M=n by MATRIX_0:24;
  thus Det RLine(M@,i,L) = Sum LaplaceExpL(RLine(M@,i,L),i) by A1,Th25
    .= Sum mlt(Col(C,i),L) by A1,A4,Th31
    .= Line(C@,i) "*" L by A1,A3,MATRIX_0:59
    .= Line(C@,i) "*" Col(M,j) by A2,A5,MATRIX_0:59;
end;
