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 Th41:
  for K being Ring
  for M1,M2 being Matrix of n,K holds
  M1 is invertible & M1 commutes_with M2 implies M1~ commutes_with M2
proof
  let K be Ring;
  let M1,M2 be Matrix of n,K;
  assume that
A1: M1 is invertible and
A2: M1 commutes_with M2;
A3: M1~ is_reverse_of M1 by A1,Def4;
A4: width M2=n by MATRIX_0:24;
A5: width M1=n & len M1=n by MATRIX_0:24;
A6: len (M2*M1)=n & width (M2*M1)=n by MATRIX_0:24;
A7: len (M1~)=n by MATRIX_0:24;
A8: len M2=n by MATRIX_0:24;
A9: width (M1~)=n by MATRIX_0:24;
  M2=(1.(K,n))*M2 by MATRIX_3:18
    .=(M1~*M1)*M2 by A3
    .=M1~*(M1*M2) by A5,A8,A9,MATRIX_3:33
    .=M1~*(M2*M1) by A2;
  then M2*M1~=M1~*((M2*M1)*M1~) by A9,A7,A6,MATRIX_3:33
    .=M1~*(M2*(M1*M1~)) by A5,A4,A7,MATRIX_3:33
    .=M1~*(M2*(1.(K,n))) by A3
    .=M1~*M2 by MATRIX_3:19;
  hence thesis;
end;
