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