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 Th35:
  Det M <> 0.K implies M~ = (Det M)" * (Matrix_of_Cofactor M)@
proof
  set C=(Det M)" * (Matrix_of_Cofactor M)@;
  assume
A1: Det M <> 0.K;
  then
A2: M*C=1.(K,n) by Th30;
  C*M=1.(K,n) by A1,Th33;
  then
A3: M is_reverse_of C by A2,MATRIX_6:def 2;
  M is invertible by A1,Th34;
  hence thesis by A3,MATRIX_6:def 4;
end;
