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 Th14:
  for R being commutative Ring, M being Matrix of n,R
  holds M is invertible implies M@ is invertible & M@~= M~@
proof
  let R be commutative Ring;
  let M be Matrix of n,R;
  assume
A1: M is invertible;
  set M1=M@, M2=M~@;
  per cases by NAT_1:3;
  suppose
A2: n > 0;
A3: width M=n & width (M~)=n by MATRIX_0:24;
  len M=n by MATRIX_0:24; then
A4: ((M~)*M)@=(M@)*(M~@) by A2,A3,MATRIX_3:22;
A5: M~ is_reverse_of M by A1,Def4; then
A6: M1*M2=1.(R,n) by A4,Th11;
  len (M~)=n by MATRIX_0:24;
  then (M*(M~))@=(M~@)*(M@) by A2,A3,MATRIX_3:22;
  then M1*M2=M2*M1 by A5,A4;
  then
A7: M1 is_reverse_of M2 by A6;
  then M1 is invertible;
  hence thesis by A7,Def4;
  end;
  suppose n = 0; then
A8: M = 1.(R,n) by MATRIX_0:45; then
A9: M1 = 1.(R,n) by Th11; then
    M@~ = 1.(R,n) by Th9;
    hence thesis by A8,A9;
  end;
end;
