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
  for R being commutative Ring, M1,M3 being Matrix of n,R
  holds M3 is_reverse_of M1 & n>0 implies M1@ is_reverse_of M3@
proof
  let R be commutative Ring;
  let M1,M3 be Matrix of n,R;
A1: width M1=n & width M3=n by MATRIX_0:24;
  assume that
A2: M3 is_reverse_of M1 and
A3: n>0;
  len M1=n & M3*M1=M1*M3 by A2,MATRIX_0:24;
  then
A4: (M1*M3)@=(M1@)*(M3@) by A1,A3,MATRIX_3:22; then
A5: (M1@)*(M3@)=1.(R,n) by Th11,A2;
  len M3=n by MATRIX_0:24;
  then (M3@)*(M1@)=(M1@)*(M3@) by A1,A3,A4,MATRIX_3:22;
  hence thesis by A5;
end;
