reserve x for set,
  D for non empty set,
  k,n,m,i,j,l for Nat,
  K for Field;

theorem Th45:
  for K being commutative Ring
  for A,B being Matrix of n,K holds Det(A*B) = (Det A)*(Det B)
proof
  let K be commutative Ring;
  let A,B be Matrix of n,K;
  per cases;
  suppose
    n > 0;
    hence thesis by MATRIX11:62;
  end;
  suppose
    n<=0;
    then
A1: n=0;
    hence Det(A*B) = 1.K by Th41
      .=1.K * 1.K
      .=(Det A)*1.K by A1,Th41
      .=(Det A)*(Det B) by A1,Th41;
  end;
end;
