reserve i,j,n for Nat,
  K for Field,
  a for Element of K,
  M,M1,M2,M3,M4 for Matrix of n,K;
reserve A for Matrix of K;

theorem Th17:
  for R being Ring, n being Nat
  for M being Matrix of n,R
  holds M is invertible implies M~ is invertible & (M~)~= M
proof
  let R be Ring, n be Nat, M be Matrix of n,R;
  assume M is invertible; then
A1: M~ is_reverse_of M by Def4;
  then M~ is invertible;
  hence thesis by A1,Def4;
end;
