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 K being Ring, M1,M2 being Matrix of n,K
  holds M2 is invertible & M2*M1=1.(K,n) implies M1 is_reverse_of M2
proof
  let K be Ring;
  let M1,M2 be Matrix of n,K;
A1: len M1=n & width M2=n by MATRIX_0:24;
A2: len M2=n & width (M2~)=n by MATRIX_0:24;
  assume that
A3: M2 is invertible and
A4: M2*M1=1.(K,n);
A5: M2~ is_reverse_of M2 by A3,Def4;
  (M2~)*(M2*M1)=M2~ by A4,MATRIX_3:19;
  then (M2~*M2)*M1=M2~ by A1,A2,MATRIX_3:33;
  then (1.(K,n))*M1=M2~ by A5;
  then M1=M2~ by MATRIX_3:18;
  hence thesis by A3,Def4;
end;
